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Properties

Label 2-624-16.5-c1-0-26
Degree $2$
Conductor $624$
Sign $0.992 + 0.126i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 − 0.626i)2-s + (−0.707 + 0.707i)3-s + (1.21 − 1.58i)4-s + (2.76 + 2.76i)5-s + (−0.453 + 1.33i)6-s − 1.16i·7-s + (0.542 − 2.77i)8-s − 1.00i·9-s + (5.23 + 1.77i)10-s + (−1.46 − 1.46i)11-s + (0.265 + 1.98i)12-s + (0.707 − 0.707i)13-s + (−0.731 − 1.47i)14-s − 3.90·15-s + (−1.05 − 3.85i)16-s + 7.07·17-s + ⋯
L(s)  = 1  + (0.896 − 0.443i)2-s + (−0.408 + 0.408i)3-s + (0.607 − 0.794i)4-s + (1.23 + 1.23i)5-s + (−0.184 + 0.546i)6-s − 0.441i·7-s + (0.191 − 0.981i)8-s − 0.333i·9-s + (1.65 + 0.560i)10-s + (−0.442 − 0.442i)11-s + (0.0766 + 0.572i)12-s + (0.196 − 0.196i)13-s + (−0.195 − 0.395i)14-s − 1.00·15-s + (−0.263 − 0.964i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.992 + 0.126i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.992 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69393 - 0.170611i\)
\(L(\frac12)\) \(\approx\) \(2.69393 - 0.170611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.626i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-2.76 - 2.76i)T + 5iT^{2} \)
7 \( 1 + 1.16iT - 7T^{2} \)
11 \( 1 + (1.46 + 1.46i)T + 11iT^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 - 8.26iT - 23T^{2} \)
29 \( 1 + (1.31 - 1.31i)T - 29iT^{2} \)
31 \( 1 - 1.74T + 31T^{2} \)
37 \( 1 + (6.81 + 6.81i)T + 37iT^{2} \)
41 \( 1 + 0.832iT - 41T^{2} \)
43 \( 1 + (4.96 + 4.96i)T + 43iT^{2} \)
47 \( 1 + 0.916T + 47T^{2} \)
53 \( 1 + (2.09 + 2.09i)T + 53iT^{2} \)
59 \( 1 + (4.76 + 4.76i)T + 59iT^{2} \)
61 \( 1 + (4.58 - 4.58i)T - 61iT^{2} \)
67 \( 1 + (-1.19 + 1.19i)T - 67iT^{2} \)
71 \( 1 + 9.14iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 9.77T + 79T^{2} \)
83 \( 1 + (-3.18 + 3.18i)T - 83iT^{2} \)
89 \( 1 + 8.57iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51480063724225732239203169263, −10.18609120549514208395906458952, −9.387025027007492017227978083258, −7.58366354715074216907645109265, −6.72792750717060744382655043771, −5.59945884398633011977534785093, −5.50491483508044850978658827912, −3.70511062088323934884430289922, −3.05131520595988098817601077386, −1.62100473816370540884049967858, 1.55458826173732537641660014345, 2.73927833785264338124415397400, 4.53548487808141011865103802277, 5.22487514643662938313008915318, 5.91875471457547101383295909301, 6.70206845464030764911384423843, 7.984968855022800514845543394069, 8.654827579148296706600709450486, 9.778973225239576069575830879913, 10.65829419011706025186964403111

Graph of the $Z$-function along the critical line