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Properties

Label 2-1150-115.114-c2-0-23
Degree $2$
Conductor $1150$
Sign $-0.742 + 0.669i$
Analytic cond. $31.3352$
Root an. cond. $5.59778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + 4.76i·3-s − 2.00·4-s − 6.73·6-s + 7.05·7-s − 2.82i·8-s − 13.6·9-s + 10.4i·11-s − 9.52i·12-s + 19.0i·13-s + 9.98i·14-s + 4.00·16-s + 12.8·17-s − 19.3i·18-s + 22.7i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.58i·3-s − 0.500·4-s − 1.12·6-s + 1.00·7-s − 0.353i·8-s − 1.52·9-s + 0.950i·11-s − 0.793i·12-s + 1.46i·13-s + 0.713i·14-s + 0.250·16-s + 0.754·17-s − 1.07i·18-s + 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(31.3352\)
Root analytic conductor: \(5.59778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1),\ -0.742 + 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.901129457\)
\(L(\frac12)\) \(\approx\) \(1.901129457\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 \)
23 \( 1 + (-22.1 + 6.14i)T \)
good3 \( 1 - 4.76iT - 9T^{2} \)
7 \( 1 - 7.05T + 49T^{2} \)
11 \( 1 - 10.4iT - 121T^{2} \)
13 \( 1 - 19.0iT - 169T^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 - 22.7iT - 361T^{2} \)
29 \( 1 - 8.61T + 841T^{2} \)
31 \( 1 - 22.2T + 961T^{2} \)
37 \( 1 + 29.8T + 1.36e3T^{2} \)
41 \( 1 + 18.7T + 1.68e3T^{2} \)
43 \( 1 + 10.0T + 1.84e3T^{2} \)
47 \( 1 + 20.6iT - 2.20e3T^{2} \)
53 \( 1 + 17.3T + 2.80e3T^{2} \)
59 \( 1 - 103.T + 3.48e3T^{2} \)
61 \( 1 - 74.6iT - 3.72e3T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + 69.4T + 5.04e3T^{2} \)
73 \( 1 + 122. iT - 5.32e3T^{2} \)
79 \( 1 + 140. iT - 6.24e3T^{2} \)
83 \( 1 - 126.T + 6.88e3T^{2} \)
89 \( 1 + 11.9iT - 7.92e3T^{2} \)
97 \( 1 - 57.0T + 9.40e3T^{2} \)
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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

−10.10485749715776778758647390691, −9.224435472086839456647282186011, −8.650084460275916869877201403604, −7.72550805239405387043306970014, −6.79128295776981506721575685558, −5.65492533752824204142057629198, −4.75535541674943353263942531598, −4.43193752246477890023757858615, −3.40020397030330829677225175515, −1.70859917215921953821725066371, 0.64333754263492691491211100031, 1.27744113971056861091839293526, 2.54738815699191744565161537952, 3.30156935523326498214505775574, 4.96319066819368956832107133434, 5.61724864650022888173876210882, 6.70417396160655226017083641466, 7.64297419394198824815268343279, 8.230824171120019694776394608654, 8.801912644354877080023559934711

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