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Properties

Label 2-1216-19.7-c1-0-14
Degree $2$
Conductor $1216$
Sign $-0.437 - 0.899i$
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  + (1.20 + 2.09i)3-s + (1.27 + 2.20i)5-s + 5.01·7-s + (−1.41 + 2.44i)9-s − 3.54·11-s + (−2.36 + 4.09i)13-s + (−3.06 + 5.31i)15-s + (−1.18 − 2.04i)17-s + (−4.31 + 0.618i)19-s + (6.04 + 10.4i)21-s + (−1.88 + 3.26i)23-s + (−0.733 + 1.27i)25-s + 0.414·27-s + (5.15 − 8.92i)29-s − 1.35·31-s + ⋯
L(s)  = 1  + (0.696 + 1.20i)3-s + (0.568 + 0.984i)5-s + 1.89·7-s + (−0.471 + 0.816i)9-s − 1.06·11-s + (−0.655 + 1.13i)13-s + (−0.792 + 1.37i)15-s + (−0.286 − 0.495i)17-s + (−0.989 + 0.142i)19-s + (1.31 + 2.28i)21-s + (−0.393 + 0.681i)23-s + (−0.146 + 0.254i)25-s + 0.0797·27-s + (0.956 − 1.65i)29-s − 0.243·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.437 - 0.899i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.528263296\)
\(L(\frac12)\) \(\approx\) \(2.528263296\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.31 - 0.618i)T \)
good3 \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.27 - 2.20i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 5.01T + 7T^{2} \)
11 \( 1 + 3.54T + 11T^{2} \)
13 \( 1 + (2.36 - 4.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.18 + 2.04i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.88 - 3.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.15 + 8.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.35T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 + (0.628 + 1.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0213 + 0.0369i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.94 + 3.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.59 + 6.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.33 + 5.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.55 - 6.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.31 - 4.01i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.967 + 1.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.11 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + (-1.64 + 2.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.343 - 0.595i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05523493619866348935599192823, −9.310413445827450776027505216708, −8.357757244249794467635537972451, −7.79519511692092810590204863054, −6.76013534763679147587797352246, −5.55152443033743866404697575503, −4.62630315367486809215052335201, −4.14787229085487391418255641769, −2.60145044449763619023899942453, −2.13156623178939323535470286391, 1.02883684095398926844137555664, 1.95086608954312125358547173191, 2.69906903723037326278924557353, 4.62371586641023977442325002798, 5.07800872789887026292003497526, 6.08875182538146990874196826026, 7.36747787225160324530780424967, 7.994796782683848543178245881099, 8.382834439281073381875022288203, 9.062069044259065827596983624392

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