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Properties

Label 2-1200-15.2-c1-0-14
Degree $2$
Conductor $1200$
Sign $0.794 - 0.607i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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  + (1.61 + 0.618i)3-s + (−1 − i)7-s + (2.23 + 2.00i)9-s + 4.47i·11-s + (3 − 3i)13-s + (2.23 − 2.23i)17-s + 2i·19-s + (−1 − 2.23i)21-s + (2.23 + 2.23i)23-s + (2.38 + 4.61i)27-s + 4.47·29-s − 4·31-s + (−2.76 + 7.23i)33-s + (3 + 3i)37-s + (6.70 − 3i)39-s + ⋯
L(s)  = 1  + (0.934 + 0.356i)3-s + (−0.377 − 0.377i)7-s + (0.745 + 0.666i)9-s + 1.34i·11-s + (0.832 − 0.832i)13-s + (0.542 − 0.542i)17-s + 0.458i·19-s + (−0.218 − 0.487i)21-s + (0.466 + 0.466i)23-s + (0.458 + 0.888i)27-s + 0.830·29-s − 0.718·31-s + (−0.481 + 1.25i)33-s + (0.493 + 0.493i)37-s + (1.07 − 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.344014308\)
\(L(\frac12)\) \(\approx\) \(2.344014308\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.61 - 0.618i)T \)
5 \( 1 \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \)
53 \( 1 + (-2.23 - 2.23i)T + 53iT^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (6.70 + 6.70i)T + 83iT^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (9 + 9i)T + 97iT^{2} \)
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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

−9.993052925237662492299096289871, −9.073494158759118018383762916384, −8.183877485838752112118797526896, −7.49075294515626963612122035538, −6.74081194355476005951239446274, −5.46573945500311285087098978571, −4.51578265455237073412710308412, −3.59706476093353242073931337957, −2.75488827550878834490373029855, −1.39990167922062572248461429956, 1.06605256918543693404033651817, 2.44212347839885635199971275619, 3.37747675322557193922622026478, 4.15613570648461564556955649550, 5.64292941542219610379464720818, 6.38830753419234978313566261659, 7.21035195870626477497172966763, 8.266791764198731151128974984147, 8.810715163518276493828971242933, 9.326812748004406622247519399746

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