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Properties

Label 2-1150-115.22-c1-0-19
Degree $2$
Conductor $1150$
Sign $0.998 + 0.0630i$
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 + (−1.22 + 1.22i)3-s + 1.00i·4-s + 1.73·6-s + (2.39 − 2.39i)7-s + (0.707 − 0.707i)8-s + 2.14i·11-s + (−1.22 − 1.22i)12-s + (−0.896 + 0.896i)13-s − 3.38·14-s − 1.00·16-s + (3.91 − 3.91i)17-s + 2.14·19-s + 5.86i·21-s + (1.51 − 1.51i)22-s + (−1.27 − 4.62i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.707 + 0.707i)3-s + 0.500i·4-s + 0.707·6-s + (0.904 − 0.904i)7-s + (0.250 − 0.250i)8-s + 0.647i·11-s + (−0.353 − 0.353i)12-s + (−0.248 + 0.248i)13-s − 0.904·14-s − 0.250·16-s + (0.948 − 0.948i)17-s + 0.492·19-s + 1.27i·21-s + (0.323 − 0.323i)22-s + (−0.266 − 0.963i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.998 + 0.0630i$
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: \((2,\ 1150,\ (\ :1/2),\ 0.998 + 0.0630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070730829\)
\(L(\frac12)\) \(\approx\) \(1.070730829\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 4.62i)T \)
good3 \( 1 + (1.22 - 1.22i)T - 3iT^{2} \)
7 \( 1 + (-2.39 + 2.39i)T - 7iT^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (0.896 - 0.896i)T - 13iT^{2} \)
17 \( 1 + (-3.91 + 3.91i)T - 17iT^{2} \)
19 \( 1 - 2.14T + 19T^{2} \)
29 \( 1 - 1.46iT - 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (1.75 - 1.75i)T - 37iT^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + (-6.54 - 6.54i)T + 43iT^{2} \)
47 \( 1 + (0.896 + 0.896i)T + 47iT^{2} \)
53 \( 1 + (1.11 + 1.11i)T + 53iT^{2} \)
59 \( 1 + 7.46iT - 59T^{2} \)
61 \( 1 - 5.86iT - 61T^{2} \)
67 \( 1 + (-0.876 + 0.876i)T - 67iT^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (-7.91 + 7.91i)T - 73iT^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \)
89 \( 1 - 9.58T + 89T^{2} \)
97 \( 1 + (6.54 - 6.54i)T - 97iT^{2} \)
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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

−9.893123020155883907959403443463, −9.354061351385631470087898059988, −8.001840143050141680407598661534, −7.59637898783304775831702243004, −6.54439483545562271983591517776, −5.11819100341591848003828414462, −4.70383400197556917575956379528, −3.72388607080748365823481287412, −2.28819611880358692568810073128, −0.889440750418556713416323581374, 0.919239604298989622892903143740, 2.02774672590713051663439319514, 3.60583974957584507783889145999, 5.20061856643288237165178580623, 5.68630528045217643382446009176, 6.35530415133415925917214900959, 7.48846527380578366640010934251, 7.998969355220534247856307843202, 8.854437569420498847463435561586, 9.665549664783783560845623456533

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