Properties

Label 2-3150-5.4-c1-0-17
Degree $2$
Conductor $3150$
Sign $0.447 - 0.894i$
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 − 2i·13-s − 14-s + 16-s − 2·19-s + 2·26-s i·28-s + 6·29-s + 8·31-s + i·32-s − 4i·37-s − 2i·38-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.458·19-s + 0.392·26-s − 0.188i·28-s + 1.11·29-s + 1.43·31-s + 0.176i·32-s − 0.657i·37-s − 0.324i·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
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: \((2,\ 3150,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.648116311\)
\(L(\frac12)\) \(\approx\) \(1.648116311\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.538725681242857869666908292632, −8.200121323913554069772071555711, −7.28002282381830107871109533855, −6.52300765544816658970992038916, −5.88210687213051791097291490817, −5.06083094473874519121372982960, −4.33758145465434388558654496862, −3.29071773038933164007453995990, −2.31612230040998051861296662072, −0.830246319784123610551317756921, 0.73863750808720447045536713662, 1.89248978017999300971331797570, 2.84891059370008941722530675145, 3.79068177135856846222756450670, 4.54944079256386358730642439459, 5.26336648189400356114605460107, 6.44244083100468576861231443861, 6.90582404766489243037836502391, 8.122393540218572032822276816006, 8.482748299997655803667908699491

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