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Properties

Label 2-950-95.8-c1-0-7
Degree $2$
Conductor $950$
Sign $-0.316 - 0.948i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.965 + 0.258i)2-s + (0.680 + 2.53i)3-s + (0.866 − 0.499i)4-s + (−1.31 − 2.27i)6-s + (2.47 − 2.47i)7-s + (−0.707 + 0.707i)8-s + (−3.38 + 1.95i)9-s + 0.295·11-s + (1.85 + 1.85i)12-s + (−1.29 − 0.347i)13-s + (−1.75 + 3.03i)14-s + (0.500 − 0.866i)16-s + (−0.182 − 0.682i)17-s + (2.76 − 2.76i)18-s + (1.91 + 3.91i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.392 + 1.46i)3-s + (0.433 − 0.249i)4-s + (−0.536 − 0.929i)6-s + (0.936 − 0.936i)7-s + (−0.249 + 0.249i)8-s + (−1.12 + 0.651i)9-s + 0.0891·11-s + (0.536 + 0.536i)12-s + (−0.360 − 0.0964i)13-s + (−0.468 + 0.810i)14-s + (0.125 − 0.216i)16-s + (−0.0443 − 0.165i)17-s + (0.651 − 0.651i)18-s + (0.440 + 0.897i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810237 + 1.12463i\)
\(L(\frac12)\) \(\approx\) \(0.810237 + 1.12463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-1.91 - 3.91i)T \)
good3 \( 1 + (-0.680 - 2.53i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.47 + 2.47i)T - 7iT^{2} \)
11 \( 1 - 0.295T + 11T^{2} \)
13 \( 1 + (1.29 + 0.347i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.182 + 0.682i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (2.00 - 7.49i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.37 - 4.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.65iT - 31T^{2} \)
37 \( 1 + (2.70 + 2.70i)T + 37iT^{2} \)
41 \( 1 + (-7.26 - 4.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.73 + 2.60i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-7.72 - 2.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.20 + 0.590i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.62 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.50 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 - 8.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.84 + 2.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.80 + 1.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.58 + 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.4 + 12.4i)T + 83iT^{2} \)
89 \( 1 + (-0.137 - 0.237i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.63 + 1.51i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.37105381359629917031256237524, −9.406157908644892787344298805074, −8.888629343217726282554581911726, −7.77705065889177357982241957598, −7.36068993079725616185782352108, −5.82901756109207650774561011090, −4.91081579867862225382016468924, −4.08062753314232347459942029086, −3.11014529255661962523646272770, −1.44611419598236948490483764425, 0.844496979032265968393523907859, 2.20593095800663779922953626950, 2.57288786430837874055313653373, 4.44100871217230193634297613936, 5.77438119508500571249050959935, 6.59443432051431672954073882074, 7.52112337585077742851994361990, 8.086015134606857292261679582968, 8.762254600423810424642048868566, 9.497616295805935481088582391975

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