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Properties

Label 2-1950-13.3-c1-0-7
Degree $2$
Conductor $1950$
Sign $-0.859 - 0.511i$
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 + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + (−2.5 + 4.33i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s − 0.999·12-s + (2.5 + 2.59i)13-s + 5·14-s + (−0.5 − 0.866i)16-s + (−4 + 6.92i)17-s + 0.999·18-s + (2.5 − 4.33i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s + (−0.944 + 1.63i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s − 0.288·12-s + (0.693 + 0.720i)13-s + 1.33·14-s + (−0.125 − 0.216i)16-s + (−0.970 + 1.68i)17-s + 0.235·18-s + (0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8817661334\)
\(L(\frac12)\) \(\approx\) \(0.8817661334\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-2.5 - 2.59i)T \)
good7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-48.5 - 84.0i)T^{2} \)
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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

−9.282476357624505982902813602347, −8.872802615036259841930815296848, −8.556830018957462287171729480271, −7.11331747970344182469529886768, −6.35288119794000054171307021912, −5.49887297664766994310760049203, −4.34516099912894322106774768519, −3.63152583939253463928614654608, −2.54627793161677866062291266947, −1.86280931083554841523579625705, 0.36417996033996846818992111353, 1.28770776188150996851293617787, 3.10113767207549748088282599526, 3.71767503134361079224763046582, 4.84756799930984082284646857199, 6.11495133337382610329286559508, 6.50574419344820441035150218202, 7.41971767222240864084276215627, 7.83540007868342819771502571880, 8.782708619609935481433878211102

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