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Properties

Label 1-760-760.507-r1-0-0
Degree $1$
Conductor $760$
Sign $0.144 + 0.989i$
Analytic cond. $81.6733$
Root an. cond. $81.6733$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.144 + 0.989i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ 0.144 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7050572728 + 0.6094494265i\)
\(L(\frac12)\) \(\approx\) \(0.7050572728 + 0.6094494265i\)
\(L(1)\) \(\approx\) \(0.6931103361 + 0.1969789036i\)
\(L(1)\) \(\approx\) \(0.6931103361 + 0.1969789036i\)

Euler product

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

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 $Z$-function along the critical line