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Properties

Label 2-950-95.13-c1-0-7
Degree $2$
Conductor $950$
Sign $0.549 - 0.835i$
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  + (−0.996 − 0.0871i)2-s + (−2.18 + 1.01i)3-s + (0.984 + 0.173i)4-s + (2.26 − 0.824i)6-s + (−0.271 − 1.01i)7-s + (−0.965 − 0.258i)8-s + (1.80 − 2.15i)9-s + (0.152 + 0.264i)11-s + (−2.32 + 0.624i)12-s + (−1.36 + 2.92i)13-s + (0.181 + 1.03i)14-s + (0.939 + 0.342i)16-s + (0.171 − 1.96i)17-s + (−1.99 + 1.99i)18-s + (1.55 − 4.07i)19-s + ⋯
L(s)  = 1  + (−0.704 − 0.0616i)2-s + (−1.26 + 0.588i)3-s + (0.492 + 0.0868i)4-s + (0.925 − 0.336i)6-s + (−0.102 − 0.382i)7-s + (−0.341 − 0.0915i)8-s + (0.603 − 0.718i)9-s + (0.0460 + 0.0797i)11-s + (−0.672 + 0.180i)12-s + (−0.378 + 0.811i)13-s + (0.0486 + 0.275i)14-s + (0.234 + 0.0855i)16-s + (0.0416 − 0.475i)17-s + (−0.469 + 0.469i)18-s + (0.355 − 0.934i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.514321 + 0.277198i\)
\(L(\frac12)\) \(\approx\) \(0.514321 + 0.277198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.996 + 0.0871i)T \)
5 \( 1 \)
19 \( 1 + (-1.55 + 4.07i)T \)
good3 \( 1 + (2.18 - 1.01i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (0.271 + 1.01i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.152 - 0.264i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.36 - 2.92i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.171 + 1.96i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-0.858 + 0.601i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (2.09 + 1.75i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.31 - 1.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.822 - 0.822i)T + 37iT^{2} \)
41 \( 1 + (2.48 - 6.83i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-3.83 + 5.48i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-0.570 + 0.0499i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-6.32 - 9.03i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (5.85 - 4.91i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.34 - 13.3i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.495 - 5.65i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (1.34 - 0.237i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.26 - 13.4i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (2.32 + 0.844i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-7.19 + 1.92i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.69 + 2.80i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-12.8 - 1.12i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.26958386078351934336220522541, −9.496963291437100802791283292050, −8.768520075714868742287904508775, −7.45828563107824889027451488506, −6.83331187572219410459364150315, −5.91536938605328487901485246938, −4.93980166786027327751228887548, −4.12819269422304467324665618382, −2.60460597977077192004552225350, −0.867865109929578207255805622196, 0.58026431728749843028122109368, 1.90691334432484768528244737614, 3.39031774265891685459897052554, 5.02700671793713827713420616279, 5.81865654404237966847055077838, 6.40453923265375761520000690374, 7.41454577080954717857084176058, 8.064790639903443123980852130596, 9.128708379021536909185047872332, 10.03792844447732505478459861955

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