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Properties

Label 2-1950-65.4-c1-0-43
Degree $2$
Conductor $1950$
Sign $0.00792 - 0.999i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + (−2.56 − 4.44i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−4.83 − 2.78i)11-s + 0.999i·12-s + (1.67 − 3.19i)13-s − 5.13·14-s + (−0.5 + 0.866i)16-s + (1.11 − 0.641i)17-s + 0.999·18-s + (−5.75 + 3.32i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.353 + 0.204i)6-s + (−0.969 − 1.67i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−1.45 − 0.841i)11-s + 0.288i·12-s + (0.465 − 0.884i)13-s − 1.37·14-s + (−0.125 + 0.216i)16-s + (0.269 − 0.155i)17-s + 0.235·18-s + (−1.32 + 0.762i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00792 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00792 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.00792 - 0.999i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.00792 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4589067135\)
\(L(\frac12)\) \(\approx\) \(0.4589067135\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-1.67 + 3.19i)T \)
good7 \( 1 + (2.56 + 4.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.83 + 2.78i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.11 + 0.641i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.75 - 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.94 - 3.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.30iT - 31T^{2} \)
37 \( 1 + (4.19 - 7.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 0.641i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.77 + 2.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.02T + 47T^{2} \)
53 \( 1 + 6.30iT - 53T^{2} \)
59 \( 1 + (1.31 - 0.759i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.19 - 8.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.43 - 2.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.44 + 5.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + (5.57 + 3.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.58 + 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.491948869995501501705695494653, −7.76791253920293614598884349857, −6.96520829221149579421199959482, −6.05448868862248270428318572309, −5.43939397919703652792983741271, −4.32782796015551220053614579343, −3.50504448781936422557518841721, −2.71728602621848592657818296353, −1.05793775283766833374277192874, −0.17846796126112086218603635251, 2.27068196708121805012969268817, 3.02022963718719308505100456462, 4.32176889949948621426564636776, 5.14593932461456808675895363361, 5.68970814476815810435656935481, 6.62651890400923602011366255025, 7.03519537292456738983534326116, 8.404894095327981359386903660793, 8.902694114632810558851108446231, 9.557453843779359615144973440428

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