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Properties

Label 2-950-95.64-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.896 + 0.442i$
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  + (−0.866 + 0.5i)2-s + (−1.35 + 0.780i)3-s + (0.499 − 0.866i)4-s + (0.780 − 1.35i)6-s + 4.56i·7-s + 0.999i·8-s + (−0.280 + 0.486i)9-s + 11-s + 1.56i·12-s + (−1.73 − i)13-s + (−2.28 − 3.95i)14-s + (−0.5 − 0.866i)16-s + (−2.70 + 1.56i)17-s − 0.561i·18-s + (2.5 + 3.57i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.780 + 0.450i)3-s + (0.249 − 0.433i)4-s + (0.318 − 0.552i)6-s + 1.72i·7-s + 0.353i·8-s + (−0.0935 + 0.162i)9-s + 0.301·11-s + 0.450i·12-s + (−0.480 − 0.277i)13-s + (−0.609 − 1.05i)14-s + (−0.125 − 0.216i)16-s + (−0.655 + 0.378i)17-s − 0.132i·18-s + (0.573 + 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106659 - 0.457677i\)
\(L(\frac12)\) \(\approx\) \(0.106659 - 0.457677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-2.5 - 3.57i)T \)
good3 \( 1 + (1.35 - 0.780i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.56iT - 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.70 - 1.56i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.65 - 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 - 7.68iT - 37T^{2} \)
41 \( 1 + (1.06 + 1.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.43 + 2.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.63 + 5.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.97 + 1.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.21 - 9.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 + 6.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.123 - 0.213i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.25 - 5.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.43 + 2.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.80iT - 83T^{2} \)
89 \( 1 + (-4.84 + 8.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.25 - 5.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.41214263692035348465786894062, −9.618102011573695954246909850970, −8.891726713541532969621774126086, −8.197563510259746473188568345819, −7.11488166844767746299534693510, −6.01884293691588451870854571441, −5.51376461489963519193607884539, −4.76920509648769849099569709147, −3.07891629297677710230774717574, −1.83752064659743478617453500853, 0.32072699169141836731112303232, 1.29059717817252957577298194546, 2.97001578291980625893395780918, 4.15055269490509394462503077305, 5.10225461351264592422387119945, 6.57362389131452272861586411292, 6.99723291374933528465523985167, 7.63377624993283709068564201434, 8.969566731772301330076814478893, 9.534935416109562428510818036462

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