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Properties

Label 2-616-616.293-c1-0-37
Degree $2$
Conductor $616$
Sign $-0.387 - 0.921i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (0.373 + 1.36i)2-s + (−1.72 + 1.01i)4-s + (2.51 − 0.817i)7-s + (−2.03 − 1.96i)8-s + (2.42 + 1.76i)9-s + (−0.0629 + 3.31i)11-s + (2.05 + 3.12i)14-s + (1.92 − 3.50i)16-s + (−1.5 + 3.96i)18-s + (−4.54 + 1.15i)22-s + 2.56·23-s + (−1.54 + 4.75i)25-s + (−3.49 + 3.96i)28-s + (2.42 + 7.45i)29-s + (5.49 + 1.32i)32-s + ⋯
L(s)  = 1  + (0.263 + 0.964i)2-s + (−0.860 + 0.508i)4-s + (0.951 − 0.309i)7-s + (−0.717 − 0.696i)8-s + (0.809 + 0.587i)9-s + (−0.0189 + 0.999i)11-s + (0.548 + 0.835i)14-s + (0.482 − 0.876i)16-s + (−0.353 + 0.935i)18-s + (−0.969 + 0.245i)22-s + 0.533·23-s + (−0.309 + 0.951i)25-s + (−0.661 + 0.749i)28-s + (0.449 + 1.38i)29-s + (0.972 + 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.387 - 0.921i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ -0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941881 + 1.41804i\)
\(L(\frac12)\) \(\approx\) \(0.941881 + 1.41804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.373 - 1.36i)T \)
7 \( 1 + (-2.51 + 0.817i)T \)
11 \( 1 + (0.0629 - 3.31i)T \)
good3 \( 1 + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 + (-2.42 - 7.45i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.31 - 0.752i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.57 + 2.17i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 + (-12.9 + 9.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.3 + 14.1i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.77885573052368283605460771185, −9.942441836276619446391895628770, −8.978813057330771108865189326372, −7.992973338576950821737614473288, −7.33543070869853787814976280105, −6.65538927799252822981238176095, −5.06520207121971035422348750121, −4.83223124665271107299847929895, −3.58201641908186390042238574561, −1.68389985787025007386994075445, 0.990409791542685417838146048598, 2.33944078580912494377969118528, 3.64060179610269429918299178252, 4.56804840202171559139586283500, 5.54870327091593477939698422022, 6.58103006831209409810172867823, 8.056969033150353651724957524405, 8.682125874204995294208065702001, 9.696563906366927948459735943434, 10.43161338235903163698585483771

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