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Properties

Label 2-975-39.29-c0-0-1
Degree $2$
Conductor $975$
Sign $0.711 + 0.702i$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + 13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.499 + 0.866i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (−1 − 1.73i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + 13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.499 + 0.866i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (−1 − 1.73i)37-s + (−0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8215877037\)
\(L(\frac12)\) \(\approx\) \(0.8215877037\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.30453205798940302719406664046, −8.972571723690020651103881054325, −8.355645739792511022006998320294, −7.47628791990059865873293477940, −6.99945707754471641375723702135, −5.85566278588905642219915718281, −4.79094098630065470339421470341, −3.91891253518455550192590590877, −2.65389537912133398228834367011, −1.02971418541924725018879198328, 1.41571598164591807696615254989, 3.16347999849495124494618806665, 4.38066645668035350937579982274, 5.08766436698850075897926323509, 5.90453157103191901404104566194, 6.45918894114801104038988629846, 8.220949813341573875702182111727, 8.719161452773775644514352834881, 9.675788111142042728602167909177, 10.18176660991191079611971912361

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