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Properties

Label 2-1950-65.9-c1-0-23
Degree $2$
Conductor $1950$
Sign $-0.957 + 0.288i$
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 + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (−2.73 + 1.58i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s − 0.999i·12-s + (−1.87 + 3.08i)13-s + 3.16·14-s + (−0.5 + 0.866i)16-s + (6.20 − 3.58i)17-s − 0.999i·18-s + (1.58 + 2.73i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (−1.03 + 0.597i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s − 0.288i·12-s + (−0.519 + 0.854i)13-s + 0.845·14-s + (−0.125 + 0.216i)16-s + (1.50 − 0.868i)17-s − 0.235i·18-s + (0.362 + 0.628i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07468793489\)
\(L(\frac12)\) \(\approx\) \(0.07468793489\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.87 - 3.08i)T \)
good7 \( 1 + (2.73 - 1.58i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-6.20 + 3.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.58 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.60 + 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.16 - 7.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + (9.08 + 5.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.74 + 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.47 - 2.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 7.16iT - 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.24 + 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.91 - 6.78i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.32iT - 73T^{2} \)
79 \( 1 + 9.16T + 79T^{2} \)
83 \( 1 - 13.6iT - 83T^{2} \)
89 \( 1 + (3.58 - 6.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.04 + 2.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.039083189573141604416614928904, −7.991503282412463660375754854637, −7.27091088488216361008704525742, −6.63915013123682400941157702680, −5.68470571534599844075669584581, −4.88436785085206757710602813783, −3.59205678070229762975642628734, −2.65777626449106021078551340781, −1.62948845732596183606672771204, −0.04144562858573414914770073029, 1.05863154599423683543359176453, 2.87202269873938630015122116637, 3.63182615586889566831606207354, 4.87196597150128535196280142884, 5.88581365115436164920982319106, 6.17396232640449795012452597444, 7.35353668223320396445201349476, 7.86715637353843204996564337490, 8.733692183430738623221564479834, 9.925859125051328948581054775395

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