Properties

Label 2-1216-152.107-c1-0-17
Degree $2$
Conductor $1216$
Sign $0.952 - 0.304i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (−0.121 + 0.0702i)3-s + (1.5 − 0.866i)5-s − 2i·7-s + (−1.49 + 2.58i)9-s + (−2.13 + 3.69i)13-s + (−0.121 + 0.210i)15-s + (1.99 + 3.44i)17-s + (4.31 − 0.630i)19-s + (0.140 + 0.243i)21-s + (6.40 + 3.70i)23-s + (−1 + 1.73i)25-s − 0.839i·27-s + (1.99 − 3.44i)29-s + 8.62·31-s + (−1.73 − 3i)35-s + ⋯
L(s)  = 1  + (−0.0702 + 0.0405i)3-s + (0.670 − 0.387i)5-s − 0.755i·7-s + (−0.496 + 0.860i)9-s + (−0.590 + 1.02i)13-s + (−0.0314 + 0.0543i)15-s + (0.482 + 0.836i)17-s + (0.989 − 0.144i)19-s + (0.0306 + 0.0530i)21-s + (1.33 + 0.771i)23-s + (−0.200 + 0.346i)25-s − 0.161i·27-s + (0.369 − 0.640i)29-s + 1.54·31-s + (−0.292 − 0.507i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771165294\)
\(L(\frac12)\) \(\approx\) \(1.771165294\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4.31 + 0.630i)T \)
good3 \( 1 + (0.121 - 0.0702i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.13 - 3.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.99 - 3.44i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.40 - 3.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.99 + 3.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 - 7.26T + 37T^{2} \)
41 \( 1 + (6.39 - 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.16 + 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.364 + 0.210i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.48 + 9.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.21 + 0.700i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.92 + 1.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.81 - 1.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.970 - 1.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.70 + 4.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.62T + 83T^{2} \)
89 \( 1 + (-14.5 - 8.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.39 - 3.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.883752618747755280848647275255, −9.033366044991932448117744193687, −8.124354918540175001196265256802, −7.32678996332363667228183958628, −6.46713278464601856286888421397, −5.37259368214409495261832921940, −4.82982791347675821124674754734, −3.65408221509146729636017539724, −2.38302742178516716849546757424, −1.20789231470629804852078465510, 0.918703440314938843646275457723, 2.78294820618134366928186452187, 2.99369698256828458574919369128, 4.76702019004959629633745607502, 5.55691225745456326643167022448, 6.26551108904555790185562777495, 7.11561017046706964309872060076, 8.135731223903664636975213340976, 8.975085072105073357555826770231, 9.741091731650082009387383335600

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