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Properties

Label 2-975-65.37-c1-0-25
Degree $2$
Conductor $975$
Sign $0.956 + 0.290i$
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  + (−1.85 + 1.07i)2-s + (−0.965 − 0.258i)3-s + (1.29 − 2.23i)4-s + (2.06 − 0.554i)6-s + (2.03 − 3.52i)7-s + 1.24i·8-s + (0.866 + 0.499i)9-s + (4.87 + 1.30i)11-s + (−1.82 + 1.82i)12-s + (−3.60 − 0.149i)13-s + 8.71i·14-s + (1.24 + 2.16i)16-s + (1.96 + 7.34i)17-s − 2.14·18-s + (−0.657 − 2.45i)19-s + ⋯
L(s)  = 1  + (−1.31 + 0.756i)2-s + (−0.557 − 0.149i)3-s + (0.645 − 1.11i)4-s + (0.844 − 0.226i)6-s + (0.769 − 1.33i)7-s + 0.440i·8-s + (0.288 + 0.166i)9-s + (1.47 + 0.394i)11-s + (−0.527 + 0.527i)12-s + (−0.999 − 0.0414i)13-s + 2.32i·14-s + (0.311 + 0.540i)16-s + (0.477 + 1.78i)17-s − 0.504·18-s + (−0.150 − 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710461 - 0.105409i\)
\(L(\frac12)\) \(\approx\) \(0.710461 - 0.105409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
13 \( 1 + (3.60 + 0.149i)T \)
good2 \( 1 + (1.85 - 1.07i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-2.03 + 3.52i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.87 - 1.30i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.96 - 7.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.657 + 2.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.00 + 7.49i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.83 + 2.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.01 + 5.01i)T + 31iT^{2} \)
37 \( 1 + (-2.64 - 4.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.523 - 1.95i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.49 + 0.400i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 2.01T + 47T^{2} \)
53 \( 1 + (5.36 - 5.36i)T - 53iT^{2} \)
59 \( 1 + (5.35 - 1.43i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.15 + 2.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.9 + 8.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.802 + 0.215i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 3.04iT - 73T^{2} \)
79 \( 1 + 4.65iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (-0.796 + 2.97i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.885 - 0.511i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.948914997879309348809507694133, −9.085822813270227427605240858365, −8.112471417715380349116520840331, −7.55042854293408688106779912676, −6.69805764277515272880682799520, −6.23607797488592248691635858772, −4.69359455554487536934361101308, −3.98580753183445897285873075136, −1.72153509945324675118965045704, −0.70826749668340742875761407783, 1.10946603112909242389369804177, 2.19331613065863758441893455905, 3.36752334975065671513974349220, 5.00185639986694708660109794050, 5.56714775548734352690768151310, 6.93657940709694980715761472070, 7.73968156922505184978845660925, 8.788951731782556752424502195031, 9.317231069443283872222582578874, 9.791246490733743892465596060759

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