Properties

Label 2-1216-152.125-c1-0-2
Degree $2$
Conductor $1216$
Sign $0.986 + 0.163i$
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  + (−2.59 − 1.5i)3-s + (−2.73 − 1.58i)5-s − 3.16·7-s + (3 + 5.19i)9-s − 3i·11-s + (−5.47 + 3.16i)13-s + (4.74 + 8.21i)15-s + (−2 + 3.46i)17-s + (−2.59 + 3.5i)19-s + (8.21 + 4.74i)21-s + (−4.74 − 8.21i)23-s + (2.5 + 4.33i)25-s − 9i·27-s + (2.73 − 1.58i)29-s − 3.16·31-s + ⋯
L(s)  = 1  + (−1.49 − 0.866i)3-s + (−1.22 − 0.707i)5-s − 1.19·7-s + (1 + 1.73i)9-s − 0.904i·11-s + (−1.51 + 0.877i)13-s + (1.22 + 2.12i)15-s + (−0.485 + 0.840i)17-s + (−0.596 + 0.802i)19-s + (1.79 + 1.03i)21-s + (−0.989 − 1.71i)23-s + (0.5 + 0.866i)25-s − 1.73i·27-s + (0.508 − 0.293i)29-s − 0.567·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09222143456\)
\(L(\frac12)\) \(\approx\) \(0.09222143456\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 3.5i)T \)
good3 \( 1 + (2.59 + 1.5i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.73 + 1.58i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (5.47 - 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.74 + 8.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.73 + 1.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 + 3.16iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.66 + 5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.58 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.06 - 3.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.73 + 1.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.33 + 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.16 - 5.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7iT - 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.933822256712994102873477338436, −8.660285327155052263623109547370, −8.011476836480493765857469815835, −6.95130561523116885571904014024, −6.49794322087648307186246261039, −5.65529738258171969722423299648, −4.59247431138898073974937916069, −3.84804938759556932956895579481, −2.12591733517300823617626605424, −0.39176378277133973314657138967, 0.13203095838916446294954479009, 2.82659163759581521881084738684, 3.77430359607055028566675913111, 4.67609688321778936454490537540, 5.35921274770510768020411900301, 6.57501756675565271496926103676, 7.01645217944862882548714563863, 7.83563527398309296722093874391, 9.488068423187399273572096884097, 9.842059213690882464767748749284

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