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Properties

Label 2-1150-115.22-c1-0-0
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 + (−0.575 + 0.575i)3-s + 1.00i·4-s + 0.814·6-s + (−2.09 + 2.09i)7-s + (0.707 − 0.707i)8-s + 2.33i·9-s + 1.39i·11-s + (−0.575 − 0.575i)12-s + (−4.24 + 4.24i)13-s + 2.96·14-s − 1.00·16-s + (4.38 − 4.38i)17-s + (1.65 − 1.65i)18-s − 2.37·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.332 + 0.332i)3-s + 0.500i·4-s + 0.332·6-s + (−0.792 + 0.792i)7-s + (0.250 − 0.250i)8-s + 0.779i·9-s + 0.422i·11-s + (−0.166 − 0.166i)12-s + (−1.17 + 1.17i)13-s + 0.792·14-s − 0.250·16-s + (1.06 − 1.06i)17-s + (0.389 − 0.389i)18-s − 0.545·19-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

\(L(1)\) \(\approx\) \(0.06054748855\)
\(L(\frac12)\) \(\approx\) \(0.06054748855\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.575 - 0.575i)T - 3iT^{2} \)
7 \( 1 + (2.09 - 2.09i)T - 7iT^{2} \)
11 \( 1 - 1.39iT - 11T^{2} \)
13 \( 1 + (4.24 - 4.24i)T - 13iT^{2} \)
17 \( 1 + (-4.38 + 4.38i)T - 17iT^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
29 \( 1 + 3.87iT - 29T^{2} \)
31 \( 1 + 5.74T + 31T^{2} \)
37 \( 1 + (-2.27 + 2.27i)T - 37iT^{2} \)
41 \( 1 - 2.49T + 41T^{2} \)
43 \( 1 + (-2.85 - 2.85i)T + 43iT^{2} \)
47 \( 1 + (-1.26 - 1.26i)T + 47iT^{2} \)
53 \( 1 + (4.13 + 4.13i)T + 53iT^{2} \)
59 \( 1 - 5.66iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 73iT^{2} \)
79 \( 1 + 5.70T + 79T^{2} \)
83 \( 1 + (11.6 + 11.6i)T + 83iT^{2} \)
89 \( 1 - 1.31T + 89T^{2} \)
97 \( 1 + (7.84 - 7.84i)T - 97iT^{2} \)
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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

−9.994885004294688535725493569682, −9.639515843415717222569592724095, −8.926673275254061951402723783057, −7.80792353255543708967621061150, −7.09739118337842542913470463125, −6.05030747438244087801561778530, −5.01095501233439682819848088507, −4.22562282240386358144049191133, −2.78931698412789977987818286025, −2.08506920143536042850020044936, 0.03430109588745761739466833721, 1.24268885709066815995861825923, 3.06964877834519220722533513409, 3.97999892384312916947884428848, 5.50882507363765938260808325892, 5.95906770233250858319075179858, 7.04175787624303133875089797175, 7.47964708588718276877054984887, 8.428356744189591795553521908955, 9.434219635360149896764130687433

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