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Properties

Label 2-1150-115.22-c1-0-13
Degree $2$
Conductor $1150$
Sign $0.932 - 0.362i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
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  + (−0.707 − 0.707i)2-s + (0.868 − 0.868i)3-s + 1.00i·4-s − 1.22·6-s + (−1.38 + 1.38i)7-s + (0.707 − 0.707i)8-s + 1.49i·9-s + 0.642i·11-s + (0.868 + 0.868i)12-s + (2.12 − 2.12i)13-s + 1.96·14-s − 1.00·16-s + (−0.903 + 0.903i)17-s + (1.05 − 1.05i)18-s − 2.55·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.501 − 0.501i)3-s + 0.500i·4-s − 0.501·6-s + (−0.525 + 0.525i)7-s + (0.250 − 0.250i)8-s + 0.497i·9-s + 0.193i·11-s + (0.250 + 0.250i)12-s + (0.589 − 0.589i)13-s + 0.525·14-s − 0.250·16-s + (−0.219 + 0.219i)17-s + (0.248 − 0.248i)18-s − 0.585·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.932 - 0.362i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 0.932 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252321684\)
\(L(\frac12)\) \(\approx\) \(1.252321684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
23 \( 1 + (-0.664 - 4.74i)T \)
good3 \( 1 + (-0.868 + 0.868i)T - 3iT^{2} \)
7 \( 1 + (1.38 - 1.38i)T - 7iT^{2} \)
11 \( 1 - 0.642iT - 11T^{2} \)
13 \( 1 + (-2.12 + 2.12i)T - 13iT^{2} \)
17 \( 1 + (0.903 - 0.903i)T - 17iT^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
29 \( 1 - 0.214iT - 29T^{2} \)
31 \( 1 - 6.15T + 31T^{2} \)
37 \( 1 + (7.20 - 7.20i)T - 37iT^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 + (-7.85 - 7.85i)T + 43iT^{2} \)
47 \( 1 + (-4.15 - 4.15i)T + 47iT^{2} \)
53 \( 1 + (-2.93 - 2.93i)T + 53iT^{2} \)
59 \( 1 + 7.08iT - 59T^{2} \)
61 \( 1 - 5.29iT - 61T^{2} \)
67 \( 1 + (-3.65 + 3.65i)T - 67iT^{2} \)
71 \( 1 + 1.83T + 71T^{2} \)
73 \( 1 + (-5.20 + 5.20i)T - 73iT^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + (6.04 + 6.04i)T + 83iT^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (10.7 - 10.7i)T - 97iT^{2} \)
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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

−9.788042384735501867013082731244, −8.975311083141651983582691114519, −8.280305369250510352839364788938, −7.64899262993838051079830890999, −6.66338060333425796962050400255, −5.73428233627931991657778971645, −4.50598227595373425490247411118, −3.24227493931035848136814603049, −2.49107800922481888036498165229, −1.33867766522752495858730049204, 0.65510549360242059568550023036, 2.41014060770720824634205567224, 3.71660534450092498697088137615, 4.35947670372490796229745801894, 5.70261505641529814765609948389, 6.62399201446663194883032098511, 7.13335527251366892233546317352, 8.470694886916701923525190601069, 8.792405457887864305621372875555, 9.638323507903147415827504333150

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