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Properties

Label 2-1950-65.49-c1-0-33
Degree $2$
Conductor $1950$
Sign $-0.712 + 0.702i$
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 + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (1.34 − 2.32i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (4.61 − 2.66i)11-s − 0.999i·12-s + (1.12 − 3.42i)13-s − 2.68·14-s + (−0.5 − 0.866i)16-s + (−3.78 − 2.18i)17-s − 0.999·18-s + (−2.70 − 1.56i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (0.508 − 0.880i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (1.39 − 0.802i)11-s − 0.288i·12-s + (0.310 − 0.950i)13-s − 0.718·14-s + (−0.125 − 0.216i)16-s + (−0.917 − 0.529i)17-s − 0.235·18-s + (−0.620 − 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.712 + 0.702i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.712 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087320715\)
\(L(\frac12)\) \(\approx\) \(1.087320715\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-1.12 + 3.42i)T \)
good7 \( 1 + (-1.34 + 2.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.61 + 2.66i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.78 + 2.18i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 + 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.828 + 0.478i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 2.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.98iT - 31T^{2} \)
37 \( 1 + (2.42 + 4.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.78 + 2.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.07 + 0.620i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.61T + 47T^{2} \)
53 \( 1 + 6.98iT - 53T^{2} \)
59 \( 1 + (5.03 + 2.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.42 - 2.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.778 + 1.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.89 - 5.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.569T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + (-0.243 + 0.140i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.01 - 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.837316235657241966164735973189, −8.466135647146242916014211725029, −7.24126885311865603686385816134, −6.66279602990356777450500233041, −5.61084391724223856657257509371, −4.58561027905890671853249546766, −3.92351769218548015909368939437, −3.00575450618258426417465036447, −1.47861248761627478182887580973, −0.51992960426285209505541033357, 1.44164312313267167284283706627, 2.20625056079292941544811885924, 4.12759565091062131330594237497, 4.58332707292724603243207986344, 5.80375027778008898301059144845, 6.36393619548141486726644213280, 6.95663213097569760266052986392, 7.923881344287740643934772879213, 8.764106874613181688194924465042, 9.220715809889082807585062762803

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