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Properties

Label 2-644-644.503-c0-0-0
Degree 22
Conductor 644644
Sign −0.552−0.833i-0.552 - 0.833i
Analytic cond. 0.3213970.321397
Root an. cond. 0.5669190.566919
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  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 + 0.540i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)9-s + (−0.118 + 0.822i)11-s + (−0.841 − 0.540i)14-s + (−0.142 − 0.989i)16-s + (−0.415 + 0.909i)18-s + (−0.797 + 0.234i)22-s + (0.654 − 0.755i)23-s + (0.142 + 0.989i)25-s + (0.142 − 0.989i)28-s + (−0.273 − 0.0801i)29-s + (0.841 − 0.540i)32-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 + 0.540i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)9-s + (−0.118 + 0.822i)11-s + (−0.841 − 0.540i)14-s + (−0.142 − 0.989i)16-s + (−0.415 + 0.909i)18-s + (−0.797 + 0.234i)22-s + (0.654 − 0.755i)23-s + (0.142 + 0.989i)25-s + (0.142 − 0.989i)28-s + (−0.273 − 0.0801i)29-s + (0.841 − 0.540i)32-s + ⋯

Functional equation

Λ(s)=(644s/2 ΓC(s) L(s)=((−0.552−0.833i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(644s/2 ΓC(s) L(s)=((−0.552−0.833i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 644644    =    22⋅7⋅232^{2} \cdot 7 \cdot 23
Sign: −0.552−0.833i-0.552 - 0.833i
Analytic conductor: 0.3213970.321397
Root analytic conductor: 0.5669190.566919
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ644(503,⋅)\chi_{644} (503, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 644, ( :0), −0.552−0.833i)(2,\ 644,\ (\ :0),\ -0.552 - 0.833i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.96920372070.9692037207
L(12)L(\frac12) ≈\approx 0.96920372070.9692037207
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)
bad2 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
7 1+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
23 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
good3 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
5 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
11 1+(0.118−0.822i)T+(−0.959−0.281i)T2 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2}
13 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
17 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
19 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
29 1+(0.273+0.0801i)T+(0.841+0.540i)T2 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2}
31 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
37 1+(−0.817+0.708i)T+(0.142−0.989i)T2 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2}
41 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
43 1+(−0.544+1.19i)T+(−0.654−0.755i)T2 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2}
47 1+T2 1 + T^{2}
53 1+(1.07+1.66i)T+(−0.415+0.909i)T2 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2}
59 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
61 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
67 1+(−0.273−1.89i)T+(−0.959+0.281i)T2 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2}
71 1+(−1.80+0.258i)T+(0.959−0.281i)T2 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2}
73 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
79 1+(0.239+0.153i)T+(0.415+0.909i)T2 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2}
83 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
89 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
97 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
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   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

−11.08385016056744693212739645328, −9.933556515823368701424538350188, −9.292890337571181266263154367733, −8.323330932532848046057096311975, −7.31262133197626150505143323109, −6.76049505301478037882514081212, −5.62177801066749771491521700016, −4.82124785268786954012190456950, −3.74698686969377715478192995498, −2.42310386482465050497776102715, 1.05746836958542969239284626486, 2.88087373688414571315150999048, 3.69370958744439064356073814291, 4.64751082407988580599695098127, 5.97110486470105728297898706401, 6.63882486976685627439062201197, 7.936482723947063389330677220462, 9.214790036010856890530396751708, 9.649651513763944191001627540665, 10.58745533598384527220885477609

Graph of the ZZ-function along the critical line