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Properties

Label 2-975-39.5-c1-0-75
Degree $2$
Conductor $975$
Sign $-0.349 + 0.936i$
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  + (1.11 + 1.11i)2-s + (−0.455 − 1.67i)3-s + 0.503i·4-s + (1.36 − 2.37i)6-s + (−1.46 − 1.46i)7-s + (1.67 − 1.67i)8-s + (−2.58 + 1.52i)9-s + (0.292 − 0.292i)11-s + (0.841 − 0.229i)12-s + (−2.19 − 2.86i)13-s − 3.28i·14-s + 4.75·16-s − 2.78·17-s + (−4.59 − 1.18i)18-s + (−1.21 + 1.21i)19-s + ⋯
L(s)  = 1  + (0.791 + 0.791i)2-s + (−0.262 − 0.964i)3-s + 0.251i·4-s + (0.555 − 0.971i)6-s + (−0.554 − 0.554i)7-s + (0.591 − 0.591i)8-s + (−0.861 + 0.507i)9-s + (0.0883 − 0.0883i)11-s + (0.242 − 0.0661i)12-s + (−0.608 − 0.793i)13-s − 0.876i·14-s + 1.18·16-s − 0.674·17-s + (−1.08 − 0.280i)18-s + (−0.277 + 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.349 + 0.936i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810189 - 1.16704i\)
\(L(\frac12)\) \(\approx\) \(0.810189 - 1.16704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.455 + 1.67i)T \)
5 \( 1 \)
13 \( 1 + (2.19 + 2.86i)T \)
good2 \( 1 + (-1.11 - 1.11i)T + 2iT^{2} \)
7 \( 1 + (1.46 + 1.46i)T + 7iT^{2} \)
11 \( 1 + (-0.292 + 0.292i)T - 11iT^{2} \)
17 \( 1 + 2.78T + 17T^{2} \)
19 \( 1 + (1.21 - 1.21i)T - 19iT^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 + (1.98 - 1.98i)T - 31iT^{2} \)
37 \( 1 + (-3.02 - 3.02i)T + 37iT^{2} \)
41 \( 1 + (3.08 + 3.08i)T + 41iT^{2} \)
43 \( 1 + 0.831iT - 43T^{2} \)
47 \( 1 + (-8.76 + 8.76i)T - 47iT^{2} \)
53 \( 1 - 0.258iT - 53T^{2} \)
59 \( 1 + (-2.38 + 2.38i)T - 59iT^{2} \)
61 \( 1 + 3.66T + 61T^{2} \)
67 \( 1 + (9.29 - 9.29i)T - 67iT^{2} \)
71 \( 1 + (7.88 + 7.88i)T + 71iT^{2} \)
73 \( 1 + (-11.5 - 11.5i)T + 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (-4.97 + 4.97i)T - 89iT^{2} \)
97 \( 1 + (-8.41 + 8.41i)T - 97iT^{2} \)
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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

−9.975546924851008428421525868960, −8.587273794316192232942986125478, −7.65286085760958197242418966139, −7.09532741553691312385320076827, −6.24094452370816521708314063102, −5.71178631904358540491996059525, −4.65284881280514618810110537760, −3.59608798313344021915560432639, −2.14299701120312016933504902644, −0.49263015635067191235445167037, 2.14972739711889505606979996941, 3.07307411838562214040878854285, 4.08720286674916397047141731348, 4.68923541758849999003974981702, 5.66166679345140983141622344683, 6.57499071569779963764294473488, 7.83894447191773025951000245335, 9.010372935244089728406700924219, 9.462180717261467275858496330497, 10.54970211632330193648316428739

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