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Properties

Label 2-3150-15.8-c1-0-13
Degree $2$
Conductor $3150$
Sign $0.999 + 0.0387i$
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 + 0.585i·11-s + (−3.43 − 3.43i)13-s + 1.00·14-s − 1.00·16-s + (0.906 + 0.906i)17-s + 2.04i·19-s + (0.414 − 0.414i)22-s + (−0.257 + 0.257i)23-s + 4.86i·26-s + (−0.707 − 0.707i)28-s + 1.75·29-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 + 0.176i·11-s + (−0.953 − 0.953i)13-s + 0.267·14-s − 0.250·16-s + (0.219 + 0.219i)17-s + 0.470i·19-s + (0.0883 − 0.0883i)22-s + (−0.0537 + 0.0537i)23-s + 0.953i·26-s + (−0.133 − 0.133i)28-s + 0.325·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0387i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (2843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.999 + 0.0387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.121564243\)
\(L(\frac12)\) \(\approx\) \(1.121564243\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 + (3.43 + 3.43i)T + 13iT^{2} \)
17 \( 1 + (-0.906 - 0.906i)T + 17iT^{2} \)
19 \( 1 - 2.04iT - 19T^{2} \)
23 \( 1 + (0.257 - 0.257i)T - 23iT^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 - 1.47T + 31T^{2} \)
37 \( 1 + (-4.04 + 4.04i)T - 37iT^{2} \)
41 \( 1 - 3.84iT - 41T^{2} \)
43 \( 1 + (5.10 + 5.10i)T + 43iT^{2} \)
47 \( 1 + (-5.49 - 5.49i)T + 47iT^{2} \)
53 \( 1 + (5.02 - 5.02i)T - 53iT^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 + (-6.74 + 6.74i)T - 67iT^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (5.97 + 5.97i)T + 73iT^{2} \)
79 \( 1 - 0.944iT - 79T^{2} \)
83 \( 1 + (-7.82 + 7.82i)T - 83iT^{2} \)
89 \( 1 - 0.0705T + 89T^{2} \)
97 \( 1 + (-0.746 + 0.746i)T - 97iT^{2} \)
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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

−8.741182750857460192288702023368, −7.941164031535419761884902130393, −7.44822949888239034101951556581, −6.47995063762519355083123325795, −5.62125663045490802222096777189, −4.78961337563564726228340486511, −3.77188498422400026478529536334, −2.89598299012985863014463256416, −2.10758038220133630552309515733, −0.77592535775976521421220301103, 0.59527336924763467924883553242, 1.94637393172584072472016517687, 2.96360891159280893940080744495, 4.15777589451826408215959423702, 4.90448563380761616764360616930, 5.73799741330026210888298053556, 6.79460264218925820319301395069, 6.98016261284486714441154568674, 7.988552528446080432047316521128, 8.607939050287288016438270917200

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