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Properties

Label 2-950-95.54-c1-0-16
Degree $2$
Conductor $950$
Sign $0.908 - 0.418i$
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  + (−0.642 + 0.766i)2-s + (0.524 + 1.43i)3-s + (−0.173 − 0.984i)4-s + (−1.43 − 0.524i)6-s + (2.33 − 1.34i)7-s + (0.866 + 0.500i)8-s + (0.5 − 0.419i)9-s + (−1.59 + 2.75i)11-s + (1.32 − 0.766i)12-s + (1.96 − 5.41i)13-s + (−0.467 + 2.65i)14-s + (−0.939 + 0.342i)16-s + (4.18 − 4.99i)17-s + 0.652i·18-s + (−2.82 − 3.31i)19-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.302 + 0.831i)3-s + (−0.0868 − 0.492i)4-s + (−0.587 − 0.213i)6-s + (0.882 − 0.509i)7-s + (0.306 + 0.176i)8-s + (0.166 − 0.139i)9-s + (−0.480 + 0.831i)11-s + (0.383 − 0.221i)12-s + (0.546 − 1.50i)13-s + (−0.125 + 0.709i)14-s + (−0.234 + 0.0855i)16-s + (1.01 − 1.21i)17-s + 0.153i·18-s + (−0.648 − 0.761i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52156 + 0.333492i\)
\(L(\frac12)\) \(\approx\) \(1.52156 + 0.333492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (2.82 + 3.31i)T \)
good3 \( 1 + (-0.524 - 1.43i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-2.33 + 1.34i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 + 5.41i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-4.18 + 4.99i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.684 - 0.120i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.16 + 1.81i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.22 + 2.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.36iT - 37T^{2} \)
41 \( 1 + (-0.326 + 0.118i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (5.97 + 1.05i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-5.06 - 6.04i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-8.08 + 1.42i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.439 + 0.368i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.509 - 2.89i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.18 + 3.79i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.46 - 8.32i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.39 - 14.8i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (8.51 - 3.10i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.34 - 4.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.27 - 2.64i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.223 + 0.266i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.10042298211461416827636644078, −9.314004517003572879668557560135, −8.371429566682867438250725470883, −7.66831232489533745896929461958, −6.98276938971561050442246532866, −5.59080491619579541224801110830, −4.88671409256266327070110410962, −4.01026635366669120478441990642, −2.69850473104890684045769270328, −0.942215992823928429196010749589, 1.44889874202413374705389921408, 2.01733698992246024177908671257, 3.40917897631649668409682777509, 4.52447482452666378032202163388, 5.78311817846065159847314308476, 6.71606245508645784256715920307, 7.78197212306849818715854793119, 8.401177259439727104532362764808, 8.808182277700715887786455918578, 10.16735673978349676911141134605

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