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Properties

Label 2-950-95.12-c1-0-22
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: $\chi_{950} (107, \cdot )$
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

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

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