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Properties

Label 2-1232-28.19-c1-0-14
Degree $2$
Conductor $1232$
Sign $0.678 - 0.734i$
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  + (0.359 + 0.622i)3-s + (1.00 + 0.579i)5-s + (2.15 + 1.53i)7-s + (1.24 − 2.15i)9-s + (0.866 − 0.5i)11-s + 6.06i·13-s + 0.832i·15-s + (5.54 − 3.20i)17-s + (−1.85 + 3.21i)19-s + (−0.181 + 1.89i)21-s + (−4.64 − 2.68i)23-s + (−1.82 − 3.16i)25-s + 3.93·27-s − 2.58·29-s + (3.48 + 6.03i)31-s + ⋯
L(s)  = 1  + (0.207 + 0.359i)3-s + (0.448 + 0.259i)5-s + (0.814 + 0.580i)7-s + (0.414 − 0.717i)9-s + (0.261 − 0.150i)11-s + 1.68i·13-s + 0.214i·15-s + (1.34 − 0.776i)17-s + (−0.426 + 0.737i)19-s + (−0.0395 + 0.412i)21-s + (−0.968 − 0.559i)23-s + (−0.365 − 0.633i)25-s + 0.758·27-s − 0.480·29-s + (0.625 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (1167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.237495785\)
\(L(\frac12)\) \(\approx\) \(2.237495785\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 1.53i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-0.359 - 0.622i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.00 - 0.579i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.06iT - 13T^{2} \)
17 \( 1 + (-5.54 + 3.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.85 - 3.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.64 + 2.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 + (-3.48 - 6.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.22 + 7.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 - 6.31iT - 43T^{2} \)
47 \( 1 + (-0.0581 + 0.100i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.656 - 1.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.31 + 2.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.60 - 1.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.60 - 1.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + (6.63 - 3.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.06 + 0.613i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.96T + 83T^{2} \)
89 \( 1 + (-9.99 - 5.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−9.757474906622338805458833227592, −9.100903511488021929570535800432, −8.356686675714629149827069905040, −7.35561089202586590624183054473, −6.41251846995789792795693901193, −5.70862208020523863135201292228, −4.53341529469116890949762560589, −3.84152660357490964712714233242, −2.50741240274936789577638803033, −1.45735310665702218372130154911, 1.09559913454588324122508252151, 2.05598993140150869365188389478, 3.39205116520682775934807541102, 4.52885639324368479971254748702, 5.39020733138842302584825849773, 6.16449644361203398737962619524, 7.63067469956583665864517435429, 7.71863517598523879504347570235, 8.569051036671944626207507260852, 9.885932375514709501989558631060

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