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Properties

Label 2-3150-5.4-c1-0-15
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 − 6·11-s − i·13-s − 14-s + 16-s + 3i·17-s + 4·19-s + 6i·22-s + 3i·23-s − 26-s + i·28-s + 3·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 1.80·11-s − 0.277i·13-s − 0.267·14-s + 0.250·16-s + 0.727i·17-s + 0.917·19-s + 1.27i·22-s + 0.625i·23-s − 0.196·26-s + 0.188i·28-s + 0.557·29-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

\(L(1)\) \(\approx\) \(1.345981977\)
\(L(\frac12)\) \(\approx\) \(1.345981977\)
\(L(\frac{3}{2})\) 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 \( 1 + 6T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.396890690561103216066728692385, −8.149724297132105758874691392591, −7.28176097123705725341777601123, −6.32047192516003649063087768939, −5.22043629582227355792129521072, −4.92710562915008985118549109562, −3.66196077915573327863344971750, −3.01192924951030813575281827571, −2.05084966473532951062737151288, −0.801377077126200488323559083344, 0.58985917366547297451182552514, 2.30977461324972319335074126966, 3.05783996250439134404285734861, 4.26477065713028948207898379877, 5.24929416754664928317637140974, 5.44546515561877733406635975891, 6.59662262362520316785158338603, 7.23548241246269076465242685135, 8.028024437350136986872283569691, 8.484416722006998457196752843412

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