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Properties

Label 2-950-95.12-c1-0-8
Degree $2$
Conductor $950$
Sign $0.971 - 0.237i$
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 + (−0.824 + 3.07i)3-s + (0.866 + 0.499i)4-s + (1.59 − 2.75i)6-s + (−3.29 − 3.29i)7-s + (−0.707 − 0.707i)8-s + (−6.18 − 3.56i)9-s + 1.07·11-s + (−2.25 + 2.25i)12-s + (0.991 − 0.265i)13-s + (2.32 + 4.03i)14-s + (0.500 + 0.866i)16-s + (−1.62 + 6.05i)17-s + (5.04 + 5.04i)18-s + (3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.475 + 1.77i)3-s + (0.433 + 0.249i)4-s + (0.649 − 1.12i)6-s + (−1.24 − 1.24i)7-s + (−0.249 − 0.249i)8-s + (−2.06 − 1.18i)9-s + 0.323·11-s + (−0.649 + 0.649i)12-s + (0.275 − 0.0737i)13-s + (0.622 + 1.07i)14-s + (0.125 + 0.216i)16-s + (−0.393 + 1.46i)17-s + (1.18 + 1.18i)18-s + (0.783 − 0.620i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.971 - 0.237i$
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: \((2,\ 950,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661829 + 0.0796681i\)
\(L(\frac12)\) \(\approx\) \(0.661829 + 0.0796681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 + 2.70i)T \)
good3 \( 1 + (0.824 - 3.07i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.29 + 3.29i)T + 7iT^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + (-0.991 + 0.265i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.62 - 6.05i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.642 + 2.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.98 + 5.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.124iT - 31T^{2} \)
37 \( 1 + (-3.61 + 3.61i)T - 37iT^{2} \)
41 \( 1 + (4.79 - 2.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.2 - 2.75i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-9.50 + 2.54i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.72 - 0.463i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.41 - 9.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.14 - 5.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.60 + 13.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.49 + 5.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.56 + 2.56i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.35 + 5.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.38 + 3.38i)T - 83iT^{2} \)
89 \( 1 + (-2.05 + 3.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.27 - 2.48i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.31728547879163016156585590139, −9.379516677850728519801492680364, −8.897490269961983563272301197810, −7.66373126429273674111985198196, −6.47638346258244785651376731484, −5.90710376234549706820736842047, −4.38745846858567969076067691217, −3.88989384299931601006243174902, −2.95842018011226391359508174449, −0.56747880286532026698291582218, 0.900576188303198063541944569516, 2.24507454283673330652378108278, 3.07800542207405777455725581844, 5.39158166254191214556141863448, 5.94775212492208141547751282758, 6.79455042162756603327666943832, 7.24898950589360369037410920588, 8.298852307207912732127821220996, 9.053683398779595833070990851438, 9.738002169624913987227972753253

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