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Properties

Label 2-1150-115.22-c1-0-10
Degree $2$
Conductor $1150$
Sign $-0.986 - 0.161i$
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 + (−0.254 + 0.254i)3-s + 1.00i·4-s − 0.360·6-s + (−0.812 + 0.812i)7-s + (−0.707 + 0.707i)8-s + 2.87i·9-s − 3.05i·11-s + (−0.254 − 0.254i)12-s + (−0.697 + 0.697i)13-s − 1.14·14-s − 1.00·16-s + (−2.89 + 2.89i)17-s + (−2.02 + 2.02i)18-s − 2.26·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.147 + 0.147i)3-s + 0.500i·4-s − 0.147·6-s + (−0.307 + 0.307i)7-s + (−0.250 + 0.250i)8-s + 0.956i·9-s − 0.920i·11-s + (−0.0735 − 0.0735i)12-s + (−0.193 + 0.193i)13-s − 0.307·14-s − 0.250·16-s + (−0.702 + 0.702i)17-s + (−0.478 + 0.478i)18-s − 0.519·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.986 - 0.161i$
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: \((2,\ 1150,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151300729\)
\(L(\frac12)\) \(\approx\) \(1.151300729\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 - 4.42i)T \)
good3 \( 1 + (0.254 - 0.254i)T - 3iT^{2} \)
7 \( 1 + (0.812 - 0.812i)T - 7iT^{2} \)
11 \( 1 + 3.05iT - 11T^{2} \)
13 \( 1 + (0.697 - 0.697i)T - 13iT^{2} \)
17 \( 1 + (2.89 - 2.89i)T - 17iT^{2} \)
19 \( 1 + 2.26T + 19T^{2} \)
29 \( 1 - 6.96iT - 29T^{2} \)
31 \( 1 + 0.938T + 31T^{2} \)
37 \( 1 + (-4.39 + 4.39i)T - 37iT^{2} \)
41 \( 1 + 3.69T + 41T^{2} \)
43 \( 1 + (2.58 + 2.58i)T + 43iT^{2} \)
47 \( 1 + (0.923 + 0.923i)T + 47iT^{2} \)
53 \( 1 + (8.88 + 8.88i)T + 53iT^{2} \)
59 \( 1 - 4.22iT - 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + (-2.79 + 2.79i)T - 67iT^{2} \)
71 \( 1 - 4.55T + 71T^{2} \)
73 \( 1 + (0.803 - 0.803i)T - 73iT^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 + (-8.37 - 8.37i)T + 83iT^{2} \)
89 \( 1 - 9.61T + 89T^{2} \)
97 \( 1 + (4.17 - 4.17i)T - 97iT^{2} \)
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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

−10.30318906377193792524310027852, −9.171784421440718999071404517322, −8.472736012962817711852938139047, −7.69726641301386700047970211068, −6.72265838125732121175770056590, −5.89835253692114438098264515807, −5.19516251639306619084799228001, −4.20784541634049102588687576820, −3.21072526924433451610156137526, −1.98523155975036123229409418223, 0.40485668785958006269189348062, 1.99740560016824840860067901975, 3.09105975406390065932590383241, 4.20450165161323239139220770516, 4.84070928151712734389841618331, 6.18930735415860527892575462132, 6.64068832654510593792600183762, 7.63267397832392318676087015261, 8.768256076810276971121840096836, 9.713526473977070787504738452932

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