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Properties

Label 2-3150-15.8-c1-0-29
Degree $2$
Conductor $3150$
Sign $-0.876 + 0.481i$
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 + 3.41i·11-s + (−1.43 − 1.43i)13-s + 1.00·14-s − 1.00·16-s + (−1.54 − 1.54i)17-s − 2.04i·19-s + (2.41 − 2.41i)22-s + (−1.15 + 1.15i)23-s + 2.03i·26-s + (−0.707 − 0.707i)28-s + 8.04·29-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 + 1.02i·11-s + (−0.399 − 0.399i)13-s + 0.267·14-s − 0.250·16-s + (−0.374 − 0.374i)17-s − 0.470i·19-s + (0.514 − 0.514i)22-s + (−0.241 + 0.241i)23-s + 0.399i·26-s + (−0.133 − 0.133i)28-s + 1.49·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.876 + 0.481i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.876 + 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4283694933\)
\(L(\frac12)\) \(\approx\) \(0.4283694933\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (1.43 + 1.43i)T + 13iT^{2} \)
17 \( 1 + (1.54 + 1.54i)T + 17iT^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 + (1.15 - 1.15i)T - 23iT^{2} \)
29 \( 1 - 8.04T + 29T^{2} \)
31 \( 1 + 6.37T + 31T^{2} \)
37 \( 1 + (-0.0498 + 0.0498i)T - 37iT^{2} \)
41 \( 1 - 7.94iT - 41T^{2} \)
43 \( 1 + (4.65 + 4.65i)T + 43iT^{2} \)
47 \( 1 + (1.39 + 1.39i)T + 47iT^{2} \)
53 \( 1 + (-2.97 + 2.97i)T - 53iT^{2} \)
59 \( 1 + 4.25T + 59T^{2} \)
61 \( 1 + 6.88T + 61T^{2} \)
67 \( 1 + (-5.84 + 5.84i)T - 67iT^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (5.97 + 5.97i)T + 73iT^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 + (1.52 - 1.52i)T - 83iT^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \)
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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

−8.486648666043558866857034624056, −7.66396667083030004675983560103, −7.02211461832187014366466963727, −6.26291142461834052789676441593, −5.09968287326740607800764297732, −4.50514821194626199178886094230, −3.36796787337537526189161239719, −2.55889692384374944975217946092, −1.65112668303865168361135161776, −0.16804208534377308460233131434, 1.14695769147468203588722410846, 2.39115011807087687853522091842, 3.50133597726714942950528179179, 4.39117815858650477910046557788, 5.36500824980677515975457444842, 6.14055254111697165725536860250, 6.74242145322393186929126203577, 7.52820787837167349354395443708, 8.332324978709654261122581904572, 8.836151615536080799999595552005

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