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Properties

Label 2-975-195.44-c1-0-63
Degree $2$
Conductor $975$
Sign $0.0775 + 0.996i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (0.789 − 0.789i)2-s + (0.548 − 1.64i)3-s + 0.754i·4-s + (−0.863 − 1.72i)6-s + (−1.97 + 1.97i)7-s + (2.17 + 2.17i)8-s + (−2.39 − 1.80i)9-s + (3.56 − 3.56i)11-s + (1.23 + 0.413i)12-s + (1.26 − 3.37i)13-s + 3.12i·14-s + 1.92·16-s − 0.700i·17-s + (−3.31 + 0.470i)18-s + (4.32 − 4.32i)19-s + ⋯
L(s)  = 1  + (0.558 − 0.558i)2-s + (0.316 − 0.948i)3-s + 0.377i·4-s + (−0.352 − 0.706i)6-s + (−0.747 + 0.747i)7-s + (0.768 + 0.768i)8-s + (−0.799 − 0.600i)9-s + (1.07 − 1.07i)11-s + (0.357 + 0.119i)12-s + (0.350 − 0.936i)13-s + 0.834i·14-s + 0.480·16-s − 0.169i·17-s + (−0.781 + 0.110i)18-s + (0.992 − 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.0775 + 0.996i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.0775 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74494 - 1.61446i\)
\(L(\frac12)\) \(\approx\) \(1.74494 - 1.61446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.548 + 1.64i)T \)
5 \( 1 \)
13 \( 1 + (-1.26 + 3.37i)T \)
good2 \( 1 + (-0.789 + 0.789i)T - 2iT^{2} \)
7 \( 1 + (1.97 - 1.97i)T - 7iT^{2} \)
11 \( 1 + (-3.56 + 3.56i)T - 11iT^{2} \)
17 \( 1 + 0.700iT - 17T^{2} \)
19 \( 1 + (-4.32 + 4.32i)T - 19iT^{2} \)
23 \( 1 - 4.02iT - 23T^{2} \)
29 \( 1 + 6.96iT - 29T^{2} \)
31 \( 1 + (-6.53 + 6.53i)T - 31iT^{2} \)
37 \( 1 + (-2.65 + 2.65i)T - 37iT^{2} \)
41 \( 1 + (-5.37 - 5.37i)T + 41iT^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 + (-3.82 - 3.82i)T + 47iT^{2} \)
53 \( 1 + 0.258T + 53T^{2} \)
59 \( 1 + (9.68 - 9.68i)T - 59iT^{2} \)
61 \( 1 + 3.98T + 61T^{2} \)
67 \( 1 + (4.24 + 4.24i)T + 67iT^{2} \)
71 \( 1 + (2.54 + 2.54i)T + 71iT^{2} \)
73 \( 1 + (6.24 - 6.24i)T - 73iT^{2} \)
79 \( 1 + 0.921T + 79T^{2} \)
83 \( 1 + (8.89 - 8.89i)T - 83iT^{2} \)
89 \( 1 + (9.13 - 9.13i)T - 89iT^{2} \)
97 \( 1 + (-1.54 - 1.54i)T + 97iT^{2} \)
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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

−9.592901824557707232619594883245, −8.972582262947460761610369065558, −8.070129450732689936920009270547, −7.41476701588479968447643657471, −6.16370770296084361362461001115, −5.73360458701811166274480139290, −4.16106928055789804295403454221, −3.04532275893910022783794812624, −2.70906124337503027520747872265, −1.02067216269233704394535975273, 1.50654095383782026848307852094, 3.33881347672401187139071244258, 4.19793565209305797520395905827, 4.72356843884385277894025968470, 5.92471535761972734786027618270, 6.72088911883192856048410609132, 7.38138528031872492209856927254, 8.751625890404181199589780025093, 9.514692025683146348759505301748, 10.12631944459611243864301639392

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