Properties

Label 2-1216-152.107-c1-0-1
Degree $2$
Conductor $1216$
Sign $-0.976 + 0.216i$
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  + (−2.52 + 1.45i)3-s + (1.80 − 1.04i)5-s + 3.63i·7-s + (2.75 − 4.76i)9-s − 3.65·11-s + (−0.883 + 1.53i)13-s + (−3.04 + 5.26i)15-s + (2.68 + 4.65i)17-s + (1.96 − 3.88i)19-s + (−5.30 − 9.19i)21-s + (−3.04 − 1.75i)23-s + (−0.324 + 0.561i)25-s + 7.29i·27-s + (−2.87 + 4.98i)29-s + 6.30·31-s + ⋯
L(s)  = 1  + (−1.45 + 0.841i)3-s + (0.807 − 0.466i)5-s + 1.37i·7-s + (0.917 − 1.58i)9-s − 1.10·11-s + (−0.244 + 0.424i)13-s + (−0.785 + 1.36i)15-s + (0.651 + 1.12i)17-s + (0.451 − 0.892i)19-s + (−1.15 − 2.00i)21-s + (−0.634 − 0.366i)23-s + (−0.0648 + 0.112i)25-s + 1.40i·27-s + (−0.534 + 0.925i)29-s + 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.976 + 0.216i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4376046768\)
\(L(\frac12)\) \(\approx\) \(0.4376046768\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 + 3.88i)T \)
good3 \( 1 + (2.52 - 1.45i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.80 + 1.04i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.63iT - 7T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + (0.883 - 1.53i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.68 - 4.65i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.04 + 1.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 - 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.30T + 31T^{2} \)
37 \( 1 + 4.58T + 37T^{2} \)
41 \( 1 + (6.75 - 3.89i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.53 - 2.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.26 - 3.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.03 + 3.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.00 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.65 + 5.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.01 + 4.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.92 + 13.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.76 + 13.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.68 + 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.97T + 83T^{2} \)
89 \( 1 + (-5.44 - 3.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.4 - 7.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34127024884174135001516685179, −9.409499112482440046359365772350, −8.905210564232834383774746500841, −7.73531493982546480910661308218, −6.30733480754119449929844432983, −5.86469156599816858221622348283, −5.14732276178892677484820597395, −4.64845093652105394641141387917, −3.07707655638291603783383721173, −1.71143301416067029206195141487, 0.22847237299047437119696275716, 1.42775280156420664520765756512, 2.77064728049306285060763203192, 4.27287812830409257273205020875, 5.50506608807169432638487674747, 5.73104160889114074114081478138, 6.90589321985407562167781689849, 7.37062505454312269503336414256, 8.044622053496248331480355394849, 9.861777450674427663406939194112

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