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Properties

Label 2-1950-65.18-c1-0-27
Degree $2$
Conductor $1950$
Sign $0.333 - 0.942i$
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  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)6-s + 4.24·7-s − i·8-s + 1.00i·9-s + (−3.39 − 3.39i)11-s + (−0.707 − 0.707i)12-s + (3.31 + 1.41i)13-s + 4.24i·14-s + 16-s + (3.15 + 3.15i)17-s − 1.00·18-s + (−1.72 − 1.72i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.288 + 0.288i)6-s + 1.60·7-s − 0.353i·8-s + 0.333i·9-s + (−1.02 − 1.02i)11-s + (−0.204 − 0.204i)12-s + (0.920 + 0.391i)13-s + 1.13i·14-s + 0.250·16-s + (0.766 + 0.766i)17-s − 0.235·18-s + (−0.395 − 0.395i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.333 - 0.942i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.361607861\)
\(L(\frac12)\) \(\approx\) \(2.361607861\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-3.31 - 1.41i)T \)
good7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + (3.39 + 3.39i)T + 11iT^{2} \)
17 \( 1 + (-3.15 - 3.15i)T + 17iT^{2} \)
19 \( 1 + (1.72 + 1.72i)T + 19iT^{2} \)
23 \( 1 + (-2.37 + 2.37i)T - 23iT^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + (-7.44 + 7.44i)T - 31iT^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + (2.65 - 2.65i)T - 41iT^{2} \)
43 \( 1 + (-4.36 + 4.36i)T - 43iT^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 + (-9.40 - 9.40i)T + 53iT^{2} \)
59 \( 1 + (2.65 - 2.65i)T - 59iT^{2} \)
61 \( 1 - 8.63T + 61T^{2} \)
67 \( 1 + 1.55iT - 67T^{2} \)
71 \( 1 + (-5.27 + 5.27i)T - 71iT^{2} \)
73 \( 1 - 9.05iT - 73T^{2} \)
79 \( 1 + 4.42iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (6.62 - 6.62i)T - 89iT^{2} \)
97 \( 1 - 6.15iT - 97T^{2} \)
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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

−8.870714433583282926364280945803, −8.464607052596420745440127126979, −8.028427068427925168237707636235, −7.15494566436340992485470807245, −5.98528113719678613983111150517, −5.35927567731529651044575747023, −4.54828772638799397780449193684, −3.72817947249876007078736781226, −2.51910266021900073042971071483, −1.13559094442926440262680661648, 1.07499203989244334646890086484, 1.97456108705127968195520357916, 2.87298991226277494648310935193, 4.00309710863465978057392724652, 4.97483228640519062106104371392, 5.48053833591971197225143558692, 6.89342662527665235104418360336, 7.78479115643909852764512487829, 8.199402605876217573941293752931, 8.899877333797216573375744036574

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