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Properties

Label 2-1950-13.4-c1-0-31
Degree $2$
Conductor $1950$
Sign $0.376 + 0.926i$
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  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (−3.96 + 2.29i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.563 − 0.325i)11-s − 0.999·12-s + (−1.31 − 3.35i)13-s − 4.58·14-s + (−0.5 + 0.866i)16-s + (−1.75 − 3.03i)17-s − 0.999i·18-s + (2.50 − 1.44i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (−1.49 + 0.865i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.169 − 0.0981i)11-s − 0.288·12-s + (−0.364 − 0.931i)13-s − 1.22·14-s + (−0.125 + 0.216i)16-s + (−0.424 − 0.735i)17-s − 0.235i·18-s + (0.575 − 0.332i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6357267378\)
\(L(\frac12)\) \(\approx\) \(0.6357267378\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.31 + 3.35i)T \)
good7 \( 1 + (3.96 - 2.29i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.563 + 0.325i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.75 + 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.50 + 1.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.42 - 2.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.78 + 8.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.58iT - 31T^{2} \)
37 \( 1 + (4.00 + 2.31i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.03 + 1.75i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.85 + 6.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.08iT - 47T^{2} \)
53 \( 1 - 5.58T + 53T^{2} \)
59 \( 1 + (-6.47 + 3.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.31 - 5.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.93 - 4.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.2 - 6.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.04iT - 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + (7.10 + 4.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.5 - 7.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.036893657454904335902883806799, −8.342727334447489119146519312966, −7.15471780530952087979856333518, −6.60411304058815181641092589851, −5.54097214076490525356566137405, −5.35745322335212465194647877299, −4.08248537329199453978212230928, −3.14215835664322825124917311099, −2.55828518564514340413343089298, −0.19415694565586654735418572341, 1.29270606902918926373492868929, 2.55681596981409516204620911669, 3.52354180911520598884776354790, 4.30309812587120931242275072988, 5.29515040877809345401916009857, 6.46324516227481173149042172613, 6.61908928835150228444804119421, 7.48069023921528651790854417593, 8.576711818584916468235161634415, 9.609335826168837412039642896292

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