Properties

Label 2-3150-5.4-c1-0-26
Degree $2$
Conductor $3150$
Sign $0.894 + 0.447i$
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  + i·2-s − 4-s i·7-s i·8-s i·13-s + 14-s + 16-s + 3i·17-s − 2·19-s − 3i·23-s + 26-s + i·28-s + 3·29-s − 31-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s − 0.277i·13-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.458·19-s − 0.625i·23-s + 0.196·26-s + 0.188i·28-s + 0.557·29-s − 0.179·31-s + 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333052697\)
\(L(\frac12)\) \(\approx\) \(1.333052697\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 15iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.514491285592630417106541837167, −7.86850025021864003922919082565, −7.11688400372109128256356569296, −6.38263968118976671161944492532, −5.73699273207765516982578236559, −4.78183526035563087960705749112, −4.09276954009409360383077629486, −3.16558000415867748122058097079, −1.88979011876853662396838087607, −0.47440218472797002822762346911, 1.06717930086113082484273531165, 2.22236095699846338261164221695, 2.99458037291442977361376496155, 3.97704346989484432646435547383, 4.78443066126930797220887524463, 5.56546816637444679235070848430, 6.43572334442883342116061027384, 7.31353699695357714789958586425, 8.143699875383429433307235246115, 8.892873197570309396618935235142

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