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Properties

Label 2-1150-115.22-c1-0-20
Degree $2$
Conductor $1150$
Sign $0.489 - 0.871i$
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 + (0.517 − 0.517i)3-s + 1.00i·4-s + 0.732·6-s + (2.78 − 2.78i)7-s + (−0.707 + 0.707i)8-s + 2.46i·9-s + 3.94i·11-s + (0.517 + 0.517i)12-s + (−3.15 + 3.15i)13-s + 3.94·14-s − 1.00·16-s + (−1.74 + 1.74i)18-s + 3.94·19-s − 2.88i·21-s + (−2.78 + 2.78i)22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.298 − 0.298i)3-s + 0.500i·4-s + 0.298·6-s + (1.05 − 1.05i)7-s + (−0.250 + 0.250i)8-s + 0.821i·9-s + 1.18i·11-s + (0.149 + 0.149i)12-s + (−0.875 + 0.875i)13-s + 1.05·14-s − 0.250·16-s + (−0.410 + 0.410i)18-s + 0.904·19-s − 0.629i·21-s + (−0.594 + 0.594i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.489 - 0.871i$
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: \((2,\ 1150,\ (\ :1/2),\ 0.489 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.563327990\)
\(L(\frac12)\) \(\approx\) \(2.563327990\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.71 - 0.855i)T \)
good3 \( 1 + (-0.517 + 0.517i)T - 3iT^{2} \)
7 \( 1 + (-2.78 + 2.78i)T - 7iT^{2} \)
11 \( 1 - 3.94iT - 11T^{2} \)
13 \( 1 + (3.15 - 3.15i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 3.94T + 19T^{2} \)
29 \( 1 + 1.19iT - 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (-7.61 + 7.61i)T - 37iT^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 + (-4.82 - 4.82i)T + 43iT^{2} \)
47 \( 1 + (-6.83 - 6.83i)T + 47iT^{2} \)
53 \( 1 + (7.61 + 7.61i)T + 53iT^{2} \)
59 \( 1 - 1.26iT - 59T^{2} \)
61 \( 1 + 2.88iT - 61T^{2} \)
67 \( 1 + (-5.57 + 5.57i)T - 67iT^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + (7.53 - 7.53i)T - 73iT^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (4.82 + 4.82i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-2.04 + 2.04i)T - 97iT^{2} \)
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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

−9.809654953330746556980349994472, −9.080382014445928793034510766613, −7.76234336795845263283325621115, −7.50015772787711586800260879866, −6.98594423808969369429510825300, −5.51792609637065469378105959127, −4.65260322817984684888790236612, −4.20520406112558853296138429615, −2.61217235491879606895495691103, −1.56472296779961077762581397239, 1.02387638948887752971806428294, 2.60624660244474541049500719820, 3.21312254546903344001931604092, 4.42433403959535496017974981748, 5.43163954088106707598203885663, 5.83735048552097588134653867932, 7.21739213899058548167212070448, 8.226001936574263336296519530849, 8.974663953283715231192143008682, 9.565033120359069429463346768622

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