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Properties

Label 2-616-11.3-c1-0-16
Degree $2$
Conductor $616$
Sign $-0.819 - 0.572i$
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  + (−0.5 − 1.53i)3-s + (−2.45 − 1.78i)5-s + (0.309 − 0.951i)7-s + (0.309 − 0.224i)9-s + (−3.31 + 0.141i)11-s + (−1.54 + 1.11i)13-s + (−1.51 + 4.66i)15-s + (−3.90 − 2.83i)17-s + (2.08 + 6.40i)19-s − 1.61·21-s + 7.99·23-s + (1.29 + 4.00i)25-s + (−4.42 − 3.21i)27-s + (0.254 − 0.782i)29-s + (−7.96 + 5.78i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.888i)3-s + (−1.09 − 0.797i)5-s + (0.116 − 0.359i)7-s + (0.103 − 0.0748i)9-s + (−0.999 + 0.0425i)11-s + (−0.427 + 0.310i)13-s + (−0.391 + 1.20i)15-s + (−0.947 − 0.688i)17-s + (0.477 + 1.46i)19-s − 0.353·21-s + 1.66·23-s + (0.259 + 0.800i)25-s + (−0.851 − 0.619i)27-s + (0.0472 − 0.145i)29-s + (−1.43 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.819 - 0.572i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ -0.819 - 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105011 + 0.333823i\)
\(L(\frac12)\) \(\approx\) \(0.105011 + 0.333823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (3.31 - 0.141i)T \)
good3 \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.45 + 1.78i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.54 - 1.11i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.90 + 2.83i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.08 - 6.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.99T + 23T^{2} \)
29 \( 1 + (-0.254 + 0.782i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.96 - 5.78i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.70 - 5.25i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.83 + 8.72i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (-1.79 - 5.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.95 + 4.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.565 - 1.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.77 + 5.64i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.23T + 67T^{2} \)
71 \( 1 + (-3.31 - 2.40i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.78 + 11.6i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.0773 - 0.0562i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.1 + 8.08i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-11.8 + 8.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.20197644753609844102452471360, −9.046373855212558988765623560517, −8.173631150258138887922878972315, −7.36559436662237646304753240294, −6.87187680137816115014622539580, −5.36853130571650135624872196246, −4.58773279446453017572606664801, −3.37094004506957232896216644328, −1.62597250112864602654819761538, −0.19810957655693543483133562654, 2.57474308921371351057231808933, 3.60799160708345667537759345483, 4.70565197717388844269929775653, 5.38348133446850940149213814908, 6.92043106421553605727541914203, 7.49435715947428966604313662013, 8.570140993762497584395396906895, 9.505676638317444792803837981019, 10.55065080998886066051409805927, 11.06461446968718422887470221001

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