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Properties

Label 2-950-95.13-c1-0-11
Degree $2$
Conductor $950$
Sign $0.711 - 0.702i$
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 + (1.70 − 0.794i)3-s + (0.984 + 0.173i)4-s + (−1.76 + 0.642i)6-s + (0.852 + 3.18i)7-s + (−0.965 − 0.258i)8-s + (0.342 − 0.407i)9-s + (0.892 + 1.54i)11-s + (1.81 − 0.486i)12-s + (0.0370 − 0.0794i)13-s + (−0.571 − 3.24i)14-s + (0.939 + 0.342i)16-s + (−0.407 + 4.66i)17-s + (−0.376 + 0.376i)18-s + (2.36 − 3.66i)19-s + ⋯
L(s)  = 1  + (−0.704 − 0.0616i)2-s + (0.983 − 0.458i)3-s + (0.492 + 0.0868i)4-s + (−0.720 + 0.262i)6-s + (0.322 + 1.20i)7-s + (−0.341 − 0.0915i)8-s + (0.114 − 0.135i)9-s + (0.269 + 0.465i)11-s + (0.524 − 0.140i)12-s + (0.0102 − 0.0220i)13-s + (−0.152 − 0.866i)14-s + (0.234 + 0.0855i)16-s + (−0.0989 + 1.13i)17-s + (−0.0886 + 0.0886i)18-s + (0.542 − 0.840i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.711 - 0.702i$
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: \((2,\ 950,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44153 + 0.591464i\)
\(L(\frac12)\) \(\approx\) \(1.44153 + 0.591464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 3.66i)T \)
good3 \( 1 + (-1.70 + 0.794i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-0.852 - 3.18i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.892 - 1.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0370 + 0.0794i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.407 - 4.66i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (3.86 - 2.70i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (1.51 + 1.27i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-6.29 - 3.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.443 - 0.443i)T + 37iT^{2} \)
41 \( 1 + (1.43 - 3.95i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.26 - 4.66i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (0.152 - 0.0133i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-2.65 - 3.78i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-10.0 + 8.40i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.37 + 13.4i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.536 + 6.13i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (2.75 - 0.485i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (3.26 + 6.99i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-12.5 - 4.55i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.98 + 1.33i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (6.11 - 2.22i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.32 - 0.728i)T + (95.5 + 16.8i)T^{2} \)
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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

−9.788694259974672776692465353394, −9.225384094686110808416931811203, −8.294049164412733600532789009858, −8.090163678458144114014039767313, −6.96971054064323417691145370103, −6.04621025436568322097484530083, −4.93654453080610550482348797232, −3.42961274457389250968247026160, −2.40866307288674449725739065458, −1.64812992366554028040534427705, 0.841708342168270259723786778811, 2.41309139903478216146052323175, 3.54388774424332567026025437567, 4.30285939525036514424078231724, 5.69131003414836058405072877346, 6.86090734483654170748295938528, 7.62797494187843419553989218891, 8.369491583644255159609380437379, 9.035146512772227431731174738233, 10.01199939542288460873906471113

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