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Properties

Label 2-975-39.17-c0-0-1
Degree $2$
Conductor $975$
Sign $0.872 + 0.488i$
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 + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 0.999·12-s − 13-s + (−0.499 + 0.866i)16-s + (−1.5 + 0.866i)19-s − 1.73i·21-s − 0.999·27-s + (1.5 + 0.866i)28-s + (0.499 − 0.866i)36-s + (−0.5 + 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + (1 − 1.73i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 0.999·12-s − 13-s + (−0.499 + 0.866i)16-s + (−1.5 + 0.866i)19-s − 1.73i·21-s − 0.999·27-s + (1.5 + 0.866i)28-s + (0.499 − 0.866i)36-s + (−0.5 + 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + (1 − 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396131347\)
\(L(\frac12)\) \(\approx\) \(1.396131347\)
\(L(1)\) 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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
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

−10.32027935987749015259357707035, −8.966118317905747823346740381909, −8.159287866113913369447574536270, −7.70645394516862979224173584284, −7.08466506440962636380442839297, −6.12580072305716025815917536416, −4.66354663951152250783043784296, −3.81787382764282396960096777260, −2.50383551609882021615672104255, −1.64178073002399830213610650935, 1.99768999552432242980904051698, 2.58471205733274075322529326526, 4.34570080172733118681798826502, 5.01961924600878079146795010603, 5.67401263155362041977398866805, 6.91944129108521930786439325482, 7.973887772583568621079626649540, 8.728029242793067043932154361261, 9.429597098470040722356123022796, 10.34380129829456399040244925527

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