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Properties

Label 2-950-19.16-c1-0-20
Degree $2$
Conductor $950$
Sign $0.757 + 0.652i$
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  + (−0.766 − 0.642i)2-s + (2.78 − 1.01i)3-s + (0.173 + 0.984i)4-s + (−2.78 − 1.01i)6-s + (0.454 + 0.787i)7-s + (0.500 − 0.866i)8-s + (4.44 − 3.72i)9-s + (−1.82 + 3.16i)11-s + (1.48 + 2.56i)12-s + (2.66 + 0.969i)13-s + (0.157 − 0.895i)14-s + (−0.939 + 0.342i)16-s + (4.13 + 3.46i)17-s − 5.80·18-s + (3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (1.60 − 0.585i)3-s + (0.0868 + 0.492i)4-s + (−1.13 − 0.414i)6-s + (0.171 + 0.297i)7-s + (0.176 − 0.306i)8-s + (1.48 − 1.24i)9-s + (−0.550 + 0.953i)11-s + (0.428 + 0.741i)12-s + (0.738 + 0.268i)13-s + (0.0422 − 0.239i)14-s + (−0.234 + 0.0855i)16-s + (1.00 + 0.840i)17-s − 1.36·18-s + (0.783 − 0.621i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10057 - 0.779427i\)
\(L(\frac12)\) \(\approx\) \(2.10057 - 0.779427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-3.41 + 2.70i)T \)
good3 \( 1 + (-2.78 + 1.01i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.454 - 0.787i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.82 - 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.66 - 0.969i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.13 - 3.46i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.226 - 1.28i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.16 + 2.65i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.98 + 5.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.82T + 37T^{2} \)
41 \( 1 + (0.0389 - 0.0141i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.54 + 8.74i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.74 + 1.46i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.76 - 10.0i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-4.30 - 3.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.646 + 3.66i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (10.4 - 8.80i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.01 + 5.75i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (13.2 - 4.81i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (14.0 - 5.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.39 + 4.15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.41 + 3.06i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-12.7 - 10.7i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.844292003211635030033660554670, −8.873213389310837981511260120331, −8.518853700005796165486186835302, −7.49488360415455379249045403504, −7.17532433309578674705441601819, −5.68299075395518133880042007724, −4.16569888942459288790784014394, −3.25613474382515821054547695915, −2.29915436051297356058315475688, −1.42623431235149342819236205965, 1.34110062495276712785930962160, 2.95367657431335127307576819518, 3.50323548129631730148986788383, 4.85152881578050831671928888127, 5.79662946113432483482474781423, 7.19560010403224000538126895919, 7.83444131146426462250756114682, 8.550259536733526592079727927211, 9.031080812845158955762950656441, 10.07141294684181082058643185638

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