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Properties

Label 2-950-95.44-c1-0-7
Degree $2$
Conductor $950$
Sign $0.950 + 0.310i$
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 + (0.291 − 0.799i)3-s + (−0.173 + 0.984i)4-s + (−0.799 + 0.291i)6-s + (−4.37 − 2.52i)7-s + (0.866 − 0.500i)8-s + (1.74 + 1.46i)9-s + (1.10 + 1.92i)11-s + (0.737 + 0.425i)12-s + (1.66 + 4.57i)13-s + (0.876 + 4.97i)14-s + (−0.939 − 0.342i)16-s + (2.81 + 3.35i)17-s − 2.27i·18-s + (3.62 + 2.42i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.168 − 0.461i)3-s + (−0.0868 + 0.492i)4-s + (−0.326 + 0.118i)6-s + (−1.65 − 0.954i)7-s + (0.306 − 0.176i)8-s + (0.580 + 0.487i)9-s + (0.334 + 0.579i)11-s + (0.212 + 0.122i)12-s + (0.462 + 1.26i)13-s + (0.234 + 1.32i)14-s + (−0.234 − 0.0855i)16-s + (0.682 + 0.812i)17-s − 0.536i·18-s + (0.831 + 0.556i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12443 - 0.178965i\)
\(L(\frac12)\) \(\approx\) \(1.12443 - 0.178965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.62 - 2.42i)T \)
good3 \( 1 + (-0.291 + 0.799i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (4.37 + 2.52i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.10 - 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.66 - 4.57i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.81 - 3.35i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (4.49 + 0.792i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.48 - 1.24i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.76 + 6.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47iT - 37T^{2} \)
41 \( 1 + (9.83 + 3.58i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-8.29 + 1.46i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.10 + 3.69i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-10.0 - 1.77i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.98 + 3.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.909 - 5.15i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.949 - 1.13i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.23 - 12.6i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (4.54 - 12.4i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.71 + 2.08i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-10.8 - 6.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.85 + 2.13i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.26 + 1.50i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.08179503358029971075010179178, −9.418448186991698819728100125589, −8.374523437886987182712391120559, −7.33815523477889157366397640097, −6.90518026770492847582423130530, −5.94505607823597861192413830905, −4.13808558228684945592114814553, −3.74898784045578346880834343844, −2.29418281375987257384302921356, −1.11506899480254043901670240843, 0.76299355100423459000204616611, 2.95241845087076542342023719881, 3.50186572971137618347094329799, 5.06382479181590121042813824061, 5.96335757782111099632633707323, 6.55499239873402977538864709578, 7.56689191325804128590022372012, 8.653070127840489502446155895196, 9.238295838429286748696654183352, 9.942778395117671425195425733655

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