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Properties

Label 2-975-195.134-c0-0-1
Degree $2$
Conductor $975$
Sign $0.561 + 0.827i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (1.5 − 0.866i)19-s − 1.73i·21-s + 0.999i·27-s + (−0.866 + 1.5i)28-s + (0.499 − 0.866i)36-s + (0.5 − 0.866i)39-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.499i)48-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (1.5 − 0.866i)19-s − 1.73i·21-s + 0.999i·27-s + (−0.866 + 1.5i)28-s + (0.499 − 0.866i)36-s + (0.5 − 0.866i)39-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.499i)48-s + (−1 + 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099363382\)
\(L(\frac12)\) \(\approx\) \(1.099363382\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−10.00210678736779568252937968940, −9.511535357476655194006887620102, −8.571615789990733212624020058553, −7.54996248150242311858563962661, −6.86924625162337166888630472066, −5.59408513026664493825857412887, −4.68796205596267753559955361145, −3.77837483024939597354319271861, −2.90823150940968889979374470656, −1.05179659945309384500839421340, 2.03520041069713283106218815118, 3.08536791602852858864582493652, 3.71451700241838555798284859208, 5.09314228891923749823928011584, 6.23634886445008342863848866109, 7.08495403209077054176760137329, 8.012982025089095922207846637588, 8.684338471496467317913400998891, 9.416807763882787432333171951251, 9.746404062137856585109115297595

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