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Properties

Label 2-1232-308.263-c1-0-8
Degree $2$
Conductor $1232$
Sign $0.811 + 0.584i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (−2.36 − 1.36i)3-s + (−0.921 − 1.59i)5-s + (−0.396 − 2.61i)7-s + (2.23 + 3.86i)9-s + (2.51 + 2.16i)11-s + 6.17i·13-s + 5.03i·15-s + (4.14 + 2.39i)17-s + (0.0381 + 0.0661i)19-s + (−2.63 + 6.73i)21-s + (2.14 − 1.24i)23-s + (0.800 − 1.38i)25-s − 4.01i·27-s + 7.13i·29-s + (−4.75 − 2.74i)31-s + ⋯
L(s)  = 1  + (−1.36 − 0.788i)3-s + (−0.412 − 0.714i)5-s + (−0.149 − 0.988i)7-s + (0.744 + 1.28i)9-s + (0.757 + 0.652i)11-s + 1.71i·13-s + 1.30i·15-s + (1.00 + 0.580i)17-s + (0.00875 + 0.0151i)19-s + (−0.575 + 1.46i)21-s + (0.448 − 0.258i)23-s + (0.160 − 0.277i)25-s − 0.771i·27-s + 1.32i·29-s + (−0.853 − 0.492i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9035081790\)
\(L(\frac12)\) \(\approx\) \(0.9035081790\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.396 + 2.61i)T \)
11 \( 1 + (-2.51 - 2.16i)T \)
good3 \( 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.921 + 1.59i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.17iT - 13T^{2} \)
17 \( 1 + (-4.14 - 2.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0381 - 0.0661i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 1.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.13iT - 29T^{2} \)
31 \( 1 + (4.75 + 2.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.83 - 6.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.30iT - 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 + (-0.725 + 0.419i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.492 + 0.852i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.20 + 1.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.49i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + (-8.99 - 5.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.83 - 3.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.42T + 83T^{2} \)
89 \( 1 + (0.870 + 1.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.56T + 97T^{2} \)
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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

−9.683568782754049657775105633278, −8.843964049331607713648402631414, −7.70773440815544767740598687225, −6.93540903568708563685660525139, −6.56251965991597287036291808868, −5.43080629540659218177102646973, −4.52310604898458788872773525341, −3.85654024845099181066805580475, −1.70309952231484024348313206703, −0.884842894773285185626511571375, 0.69277277062926079834053643476, 2.89888450996068346491898139969, 3.63124326680035281239452339688, 4.88815130821970384877850586380, 5.78527050862355056509587255296, 5.99891697396125486716308541375, 7.23770342732502370662595549054, 8.114068131765848164731310791654, 9.290217701369032124043740411170, 9.865097686111367599590225380652

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