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Properties

Label 2-1950-13.10-c1-0-34
Degree $2$
Conductor $1950$
Sign $-0.751 + 0.659i$
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 + (−1.32 − 0.763i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.14 + 0.658i)11-s − 0.999·12-s + (2.67 − 2.41i)13-s + 1.52·14-s + (−0.5 − 0.866i)16-s + (−0.784 + 1.35i)17-s − 0.999i·18-s + (4.18 + 2.41i)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 + (−0.500 − 0.288i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.343 + 0.198i)11-s − 0.288·12-s + (0.743 − 0.669i)13-s + 0.408·14-s + (−0.125 − 0.216i)16-s + (−0.190 + 0.329i)17-s − 0.235i·18-s + (0.959 + 0.553i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4935323392\)
\(L(\frac12)\) \(\approx\) \(0.4935323392\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.67 + 2.41i)T \)
good7 \( 1 + (1.32 + 0.763i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.14 - 0.658i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.784 - 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.18 - 2.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.22 + 7.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.21 - 3.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.62iT - 31T^{2} \)
37 \( 1 + (2.42 - 1.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.35 + 0.784i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.64 + 4.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.94iT - 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
59 \( 1 + (-9.07 - 5.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.40 + 1.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.8 + 7.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.98iT - 73T^{2} \)
79 \( 1 + 4.87T + 79T^{2} \)
83 \( 1 + 6.39iT - 83T^{2} \)
89 \( 1 + (15.9 - 9.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.66 + 0.963i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.628658253992342948211797382525, −8.154771134847125840208055156393, −7.32335448526969181986930158810, −6.57659405865554540137263644141, −5.92497911980940088043777978430, −5.09974371165381895101661458125, −3.85354424168792672806475181604, −2.73476170753788469813270830792, −1.47922806319568926932738150451, −0.24427766676737646191067738040, 1.33205265666439120279945328735, 2.71523255123121203355872973290, 3.53194461688700703032683862332, 4.49335215190845673565458113307, 5.58948596747402368692152928213, 6.28978371961392056977081680279, 7.23359271962557119159219505252, 8.062689200777464119057775041644, 8.940848019204005934298524757876, 9.585008789435444981498613204117

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