Properties

Label 2-1216-152.45-c1-0-23
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: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.518956216\)
\(L(\frac12)\) \(\approx\) \(1.518956216\)
\(L(\frac{3}{2})\) 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

−9.797924119813647728001657034482, −9.126644583561301454893592462734, −8.452636364464732509657817661592, −6.79712174932698569176306669089, −6.33935823444299934566915093774, −5.28089465823134409439241386569, −4.76574887041001768036612311776, −4.25675835446787450401894313850, −2.12456030084789774003236993866, −0.991161870433642469901843128828, 1.21968391557942937004785753262, 1.85401232129347677795160331513, 3.49840334045745115838349420947, 5.05442053662782543938097290932, 5.86344468794532499006753932281, 6.03315559530527108263681074386, 6.98601006076375700834915627108, 7.989914647657732317122082123375, 8.703864183413710063839506201465, 10.12622616702188260625299834180

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