Properties

Label 1-760-760.507-r1-0-0
Degree 11
Conductor 760760
Sign 0.144+0.989i0.144 + 0.989i
Analytic cond. 81.673381.6733
Root an. cond. 81.673381.6733
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)3-s + (−0.866 + 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (0.984 + 0.173i)17-s + (0.173 − 0.984i)21-s + (0.342 − 0.939i)23-s + (0.866 + 0.5i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.342 − 0.939i)33-s i·37-s + 39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (−0.866 + 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (0.984 + 0.173i)17-s + (0.173 − 0.984i)21-s + (0.342 − 0.939i)23-s + (0.866 + 0.5i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.342 − 0.939i)33-s i·37-s + 39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

Λ(s)=(760s/2ΓR(s+1)L(s)=((0.144+0.989i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(760s/2ΓR(s+1)L(s)=((0.144+0.989i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 760760    =    235192^{3} \cdot 5 \cdot 19
Sign: 0.144+0.989i0.144 + 0.989i
Analytic conductor: 81.673381.6733
Root analytic conductor: 81.673381.6733
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ760(507,)\chi_{760} (507, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 760, (1: ), 0.144+0.989i)(1,\ 760,\ (1:\ ),\ 0.144 + 0.989i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.7050572728+0.6094494265i0.7050572728 + 0.6094494265i
L(12)L(\frac12) \approx 0.7050572728+0.6094494265i0.7050572728 + 0.6094494265i
L(1)L(1) \approx 0.6931103361+0.1969789036i0.6931103361 + 0.1969789036i
L(1)L(1) \approx 0.6931103361+0.1969789036i0.6931103361 + 0.1969789036i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
19 1 1
good3 1+(0.642+0.766i)T 1 + (-0.642 + 0.766i)T
7 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
11 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
17 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
23 1+(0.3420.939i)T 1 + (0.342 - 0.939i)T
29 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
31 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
37 1iT 1 - iT
41 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
43 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
47 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
53 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
59 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
61 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
67 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
71 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
73 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
79 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
83 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
89 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
97 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.00715143128472983606376136963, −21.50410165913331622882384928561, −20.25668469478385691558240175587, −19.285397133716598698035366402309, −18.97450412643259099976581479991, −18.03748346818140111562148906565, −17.050613571800287109129060093426, −16.474226496188999507738911267673, −15.86465687309798473772529979422, −14.384962208009346726123976441659, −13.744394354540920531580903982383, −12.90167773782433141550637788505, −12.23639674823704429009318133649, −11.29907438603118104489348299755, −10.5169935212600565224452851416, −9.597574864680680455839801741852, −8.50415636884229109013056425821, −7.32352043331475066342246215630, −6.958717420693916807684977315164, −5.78311924396985953338203277918, −5.16849115120727584489609864884, −3.73663690744993946731850964611, −2.7650811244468512912135899093, −1.45940890337953272252140861622, −0.3961051783918262991603893880, 0.61636821450009234788703199225, 2.403966318335426569445511815590, 3.31185027261953433915891907732, 4.41699116738224151582095968519, 5.33601453742702006417496241342, 6.02773611149079249305407929545, 7.06074140739109549116875860825, 8.12967487524772643984599276989, 9.3544305819126467453313565065, 9.96235049127422260692518607795, 10.53880101126422050178123154105, 11.7602069114106258593131549518, 12.45167626512931117757227803013, 13.06891492796902351219936621296, 14.5486171268066207308080809205, 15.2326862078895134575819858050, 15.79291414029997213711893382055, 16.800586038723304611840262628352, 17.30283084689924671593180346603, 18.37591727984166005954938931544, 19.033566252432319826723607693765, 20.270513244710874082406794222162, 20.72496511448246537766759477337, 21.77396255369341927171547313163, 22.43715874612859759405615120927

Graph of the ZZ-function along the critical line