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Properties

Label 2-624-16.13-c1-0-46
Degree $2$
Conductor $624$
Sign $-0.0209 + 0.999i$
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  + (1.09 − 0.891i)2-s + (0.707 + 0.707i)3-s + (0.410 − 1.95i)4-s + (0.927 − 0.927i)5-s + (1.40 + 0.145i)6-s − 3.33i·7-s + (−1.29 − 2.51i)8-s + 1.00i·9-s + (0.191 − 1.84i)10-s + (−0.615 + 0.615i)11-s + (1.67 − 1.09i)12-s + (−0.707 − 0.707i)13-s + (−2.97 − 3.65i)14-s + 1.31·15-s + (−3.66 − 1.60i)16-s − 2.54·17-s + ⋯
L(s)  = 1  + (0.776 − 0.630i)2-s + (0.408 + 0.408i)3-s + (0.205 − 0.978i)4-s + (0.414 − 0.414i)5-s + (0.574 + 0.0596i)6-s − 1.25i·7-s + (−0.457 − 0.889i)8-s + 0.333i·9-s + (0.0605 − 0.583i)10-s + (−0.185 + 0.185i)11-s + (0.483 − 0.315i)12-s + (−0.196 − 0.196i)13-s + (−0.793 − 0.977i)14-s + 0.338·15-s + (−0.915 − 0.401i)16-s − 0.616·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0209 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0209 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.0209 + 0.999i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.0209 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83121 - 1.87002i\)
\(L(\frac12)\) \(\approx\) \(1.83121 - 1.87002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.891i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-0.927 + 0.927i)T - 5iT^{2} \)
7 \( 1 + 3.33iT - 7T^{2} \)
11 \( 1 + (0.615 - 0.615i)T - 11iT^{2} \)
17 \( 1 + 2.54T + 17T^{2} \)
19 \( 1 + (-3.03 - 3.03i)T + 19iT^{2} \)
23 \( 1 + 3.71iT - 23T^{2} \)
29 \( 1 + (-5.22 - 5.22i)T + 29iT^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + (-3.24 + 3.24i)T - 37iT^{2} \)
41 \( 1 - 5.26iT - 41T^{2} \)
43 \( 1 + (-5.05 + 5.05i)T - 43iT^{2} \)
47 \( 1 - 8.90T + 47T^{2} \)
53 \( 1 + (4.32 - 4.32i)T - 53iT^{2} \)
59 \( 1 + (-0.909 + 0.909i)T - 59iT^{2} \)
61 \( 1 + (2.78 + 2.78i)T + 61iT^{2} \)
67 \( 1 + (-5.12 - 5.12i)T + 67iT^{2} \)
71 \( 1 - 5.46iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 4.95T + 79T^{2} \)
83 \( 1 + (0.643 + 0.643i)T + 83iT^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−10.48115497218252325708533301124, −9.773205121733935484551252850186, −8.946745164982070600823038445012, −7.66439442780433423421054440181, −6.70658540661717540744332674759, −5.49547440888572792639376768270, −4.59803585995498224818314722378, −3.81608825688011289432687894043, −2.63792693999506314501671961520, −1.17463364133031863293460894243, 2.33071642399203817015907388376, 2.92886751320599432569864313542, 4.42118397058786587599033738733, 5.56321499748526781894611182765, 6.25610638889905392223009448496, 7.12480263748619491568195545125, 8.107033315115778795316827829429, 8.877498658384635439857884480591, 9.712171215012926506426163222982, 11.14624139994580342006199927537

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