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Properties

Label 2-975-39.8-c1-0-43
Degree $2$
Conductor $975$
Sign $0.415 + 0.909i$
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  + (−0.260 + 0.260i)2-s + (−1.18 + 1.26i)3-s + 1.86i·4-s + (−0.0199 − 0.637i)6-s + (−2.54 + 2.54i)7-s + (−1.00 − 1.00i)8-s + (−0.187 − 2.99i)9-s + (0.348 + 0.348i)11-s + (−2.35 − 2.21i)12-s + (0.339 − 3.58i)13-s − 1.32i·14-s − 3.20·16-s − 5.28·17-s + (0.828 + 0.730i)18-s + (−3.44 − 3.44i)19-s + ⋯
L(s)  = 1  + (−0.184 + 0.184i)2-s + (−0.684 + 0.728i)3-s + 0.932i·4-s + (−0.00814 − 0.260i)6-s + (−0.961 + 0.961i)7-s + (−0.355 − 0.355i)8-s + (−0.0625 − 0.998i)9-s + (0.105 + 0.105i)11-s + (−0.679 − 0.638i)12-s + (0.0942 − 0.995i)13-s − 0.354i·14-s − 0.801·16-s − 1.28·17-s + (0.195 + 0.172i)18-s + (−0.791 − 0.791i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (476, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104594 - 0.0672291i\)
\(L(\frac12)\) \(\approx\) \(0.104594 - 0.0672291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 - 1.26i)T \)
5 \( 1 \)
13 \( 1 + (-0.339 + 3.58i)T \)
good2 \( 1 + (0.260 - 0.260i)T - 2iT^{2} \)
7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 + (-0.348 - 0.348i)T + 11iT^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + (3.44 + 3.44i)T + 19iT^{2} \)
23 \( 1 - 9.38T + 23T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (2.96 + 2.96i)T + 31iT^{2} \)
37 \( 1 + (1.76 - 1.76i)T - 37iT^{2} \)
41 \( 1 + (2.21 - 2.21i)T - 41iT^{2} \)
43 \( 1 + 5.41iT - 43T^{2} \)
47 \( 1 + (-2.46 - 2.46i)T + 47iT^{2} \)
53 \( 1 + 6.94iT - 53T^{2} \)
59 \( 1 + (0.248 + 0.248i)T + 59iT^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 + (-1.34 - 1.34i)T + 67iT^{2} \)
71 \( 1 + (-11.6 + 11.6i)T - 71iT^{2} \)
73 \( 1 + (6.48 - 6.48i)T - 73iT^{2} \)
79 \( 1 - 2.66T + 79T^{2} \)
83 \( 1 + (7.35 - 7.35i)T - 83iT^{2} \)
89 \( 1 + (2.54 + 2.54i)T + 89iT^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−9.664593150521333944260219303623, −8.947009499582204876300143253986, −8.584952140663050932737988854710, −7.02369778585066386066836262774, −6.59094759363418719280014902429, −5.50737696677903670149266185157, −4.60650384057287383219543433417, −3.43186910829931640712403868000, −2.71143564503179442209217195965, −0.07181608987334717141490034342, 1.21069391289839799204592733696, 2.39133608913741070864920606797, 4.04820037614769437658790350546, 5.00246433092549571199761791691, 6.19094192010283860202385695093, 6.62688232052357376217393993908, 7.29259536765145617098042452558, 8.675580925614239695178697670366, 9.408529980008734844705628936279, 10.37553084003700731852128210303

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