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Properties

Label 2-3150-15.2-c1-0-10
Degree $2$
Conductor $3150$
Sign $0.920 - 0.391i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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 − 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − 0.828i·11-s + (−0.585 + 0.585i)13-s − 1.00·14-s − 1.00·16-s + (−4.41 + 4.41i)17-s + 7.07i·19-s + (−0.585 − 0.585i)22-s + (0.828 + 0.828i)23-s + 0.828i·26-s + (−0.707 + 0.707i)28-s + 8.82·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s − 0.249i·11-s + (−0.162 + 0.162i)13-s − 0.267·14-s − 0.250·16-s + (−1.07 + 1.07i)17-s + 1.62i·19-s + (−0.124 − 0.124i)22-s + (0.172 + 0.172i)23-s + 0.162i·26-s + (−0.133 + 0.133i)28-s + 1.63·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.849962593\)
\(L(\frac12)\) \(\approx\) \(1.849962593\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (0.585 - 0.585i)T - 13iT^{2} \)
17 \( 1 + (4.41 - 4.41i)T - 17iT^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 + (-0.828 - 0.828i)T + 23iT^{2} \)
29 \( 1 - 8.82T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-8.07 - 8.07i)T + 37iT^{2} \)
41 \( 1 - 0.828iT - 41T^{2} \)
43 \( 1 + (-3.58 + 3.58i)T - 43iT^{2} \)
47 \( 1 + (3.58 - 3.58i)T - 47iT^{2} \)
53 \( 1 + (0.828 + 0.828i)T + 53iT^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + (2.41 + 2.41i)T + 67iT^{2} \)
71 \( 1 + 3.75iT - 71T^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 - 9.31iT - 79T^{2} \)
83 \( 1 + (1.65 + 1.65i)T + 83iT^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + (-0.242 - 0.242i)T + 97iT^{2} \)
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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

−8.626401710329880428236030551693, −8.169355393188612734992913253779, −7.06829328124899939097360243426, −6.30597431908173016635398092868, −5.75800527226934302478446549260, −4.65176934704160784243273241255, −4.03124987318200137611955783971, −3.20367700118773287726673881381, −2.18883017624833396015244947753, −1.14598137109386565445080191204, 0.51920011179929926720435907178, 2.39328013354049563204920864410, 2.89271933453093806243905262050, 4.18556990125430740096454471695, 4.78738224738687406590127760567, 5.51710208787576701465028035149, 6.53085411447327993439922401296, 6.96143340519452991967018114441, 7.70437195534241018625052617316, 8.720556899339470689910797065539

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