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Properties

Label 2-975-65.29-c1-0-6
Degree $2$
Conductor $975$
Sign $-0.636 - 0.771i$
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  + (0.294 − 0.169i)2-s + (−0.866 + 0.5i)3-s + (−0.942 + 1.63i)4-s + (−0.169 + 0.294i)6-s + (0.571 + 0.330i)7-s + 1.32i·8-s + (0.499 − 0.866i)9-s + (−0.339 − 0.588i)11-s − 1.88i·12-s + (3.04 + 1.93i)13-s + 0.224·14-s + (−1.66 − 2.87i)16-s + (6.43 + 3.71i)17-s − 0.339i·18-s + (0.0577 − 0.100i)19-s + ⋯
L(s)  = 1  + (0.208 − 0.120i)2-s + (−0.499 + 0.288i)3-s + (−0.471 + 0.816i)4-s + (−0.0693 + 0.120i)6-s + (0.216 + 0.124i)7-s + 0.466i·8-s + (0.166 − 0.288i)9-s + (−0.102 − 0.177i)11-s − 0.544i·12-s + (0.844 + 0.535i)13-s + 0.0599·14-s + (−0.415 − 0.718i)16-s + (1.56 + 0.900i)17-s − 0.0801i·18-s + (0.0132 − 0.0229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446524 + 0.947271i\)
\(L(\frac12)\) \(\approx\) \(0.446524 + 0.947271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.04 - 1.93i)T \)
good2 \( 1 + (-0.294 + 0.169i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.571 - 0.330i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.339 + 0.588i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.43 - 3.71i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0577 + 0.100i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.72 - 3.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.77 - 4.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.97T + 31T^{2} \)
37 \( 1 + (8.46 - 4.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.11 + 3.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 - 0.272i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.01iT - 47T^{2} \)
53 \( 1 - 0.679iT - 53T^{2} \)
59 \( 1 + (-1.11 + 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.10 + 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.61 + 3.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.65 - 6.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.01iT - 73T^{2} \)
79 \( 1 + 9.97T + 79T^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 + (-6.77 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 4.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36523952006192471928985403839, −9.482112233410213201585865045491, −8.560267677651457704687144467723, −7.962846220169560381779026001969, −6.93570240812985865432807607550, −5.76814657728264851708276153531, −5.13276352085461454864687226097, −3.85362326806452521916580067135, −3.45260880224225287836742456958, −1.65974582626602317774906310468, 0.50784747088814196214706363586, 1.77802606313867218199500113497, 3.49818764229949516391169914234, 4.57809472537061252924314484243, 5.54519336329690893624131925688, 5.97534996213212904326652460063, 7.09358317549898453552414621904, 7.959730160480760034253498574598, 8.901593078730914455637320355104, 9.976390474445808657493279182123

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