↑

Properties

Label 2-1035-115.34-c0-0-1
Degree 22
Conductor 10351035
Sign −0.874−0.484i-0.874 - 0.484i
Analytic cond. 0.5165320.516532
Root an. cond. 0.7187010.718701
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 1.66i)2-s + (−1.21 + 2.65i)4-s + (−0.959 + 0.281i)5-s + (3.76 − 0.540i)8-s + (1.49 + 1.29i)10-s + (−3.01 − 3.47i)16-s + (−0.345 − 0.755i)17-s + (−1.37 − 0.627i)19-s + (0.415 − 2.88i)20-s + (−0.841 − 0.540i)23-s + (0.841 − 0.540i)25-s + (−0.186 − 1.29i)31-s + (−1.49 + 5.09i)32-s + (−0.889 + 1.38i)34-s + (0.425 + 2.96i)38-s + ⋯
L(s)  = 1  + (−1.07 − 1.66i)2-s + (−1.21 + 2.65i)4-s + (−0.959 + 0.281i)5-s + (3.76 − 0.540i)8-s + (1.49 + 1.29i)10-s + (−3.01 − 3.47i)16-s + (−0.345 − 0.755i)17-s + (−1.37 − 0.627i)19-s + (0.415 − 2.88i)20-s + (−0.841 − 0.540i)23-s + (0.841 − 0.540i)25-s + (−0.186 − 1.29i)31-s + (−1.49 + 5.09i)32-s + (−0.889 + 1.38i)34-s + (0.425 + 2.96i)38-s + ⋯

Functional equation

Λ(s)=(1035s/2 ΓC(s) L(s)=((−0.874−0.484i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1035s/2 ΓC(s) L(s)=((−0.874−0.484i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10351035    =    32⋅5⋅233^{2} \cdot 5 \cdot 23
Sign: −0.874−0.484i-0.874 - 0.484i
Analytic conductor: 0.5165320.516532
Root analytic conductor: 0.7187010.718701
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1035(379,⋅)\chi_{1035} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1035, ( :0), −0.874−0.484i)(2,\ 1035,\ (\ :0),\ -0.874 - 0.484i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.21845840800.2184584080
L(12)L(\frac12) ≈\approx 0.21845840800.2184584080
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1+(0.959−0.281i)T 1 + (0.959 - 0.281i)T
23 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
good2 1+(1.07+1.66i)T+(−0.415+0.909i)T2 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2}
7 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
11 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
13 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
17 1+(0.345+0.755i)T+(−0.654+0.755i)T2 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2}
19 1+(1.37+0.627i)T+(0.654+0.755i)T2 1 + (1.37 + 0.627i)T + (0.654 + 0.755i)T^{2}
29 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
31 1+(0.186+1.29i)T+(−0.959+0.281i)T2 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2}
37 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
41 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
43 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
47 1−1.08iT−T2 1 - 1.08iT - T^{2}
53 1+(1.10+1.27i)T+(−0.142+0.989i)T2 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2}
59 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
61 1+(1.07−0.153i)T+(0.959−0.281i)T2 1 + (1.07 - 0.153i)T + (0.959 - 0.281i)T^{2}
67 1+(0.415−0.909i)T2 1 + (0.415 - 0.909i)T^{2}
71 1+(0.415−0.909i)T2 1 + (0.415 - 0.909i)T^{2}
73 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
79 1+(0.425+0.368i)T+(0.142+0.989i)T2 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2}
83 1+(−0.273−0.0801i)T+(0.841+0.540i)T2 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2}
89 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
97 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.774050379559264785778925874880, −8.959927304988823753249698351443, −8.253161014959665000797905502987, −7.61113835658180741986047820332, −6.65688751056734141560086723345, −4.63241848854074474056744820023, −4.03895152004890064045443059973, −2.98076015810112659231509884997, −2.08939265985803490993156570244, −0.29610066118754624069314019043, 1.57706190386261152032571997430, 3.97386807488616327472987822238, 4.77357240098153922524798364969, 5.84583493832856951881473504578, 6.56521378366491621612206425272, 7.44202184236778392749718721948, 8.140057810051660764589355190455, 8.639073411155034599709506349032, 9.405385474724249998102063936276, 10.47752249866333772565391604694

Graph of the ZZ-function along the critical line