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Properties

Label 2-950-95.44-c1-0-24
Degree $2$
Conductor $950$
Sign $-0.995 + 0.0981i$
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 + (0.412 − 1.13i)3-s + (−0.173 + 0.984i)4-s + (−1.13 + 0.412i)6-s + (−1.90 − 1.09i)7-s + (0.866 − 0.500i)8-s + (1.18 + 0.992i)9-s + (0.158 + 0.274i)11-s + (1.04 + 0.603i)12-s + (−0.0234 − 0.0644i)13-s + (0.381 + 2.16i)14-s + (−0.939 − 0.342i)16-s + (−4.30 − 5.12i)17-s − 1.54i·18-s + (0.761 − 4.29i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.238 − 0.654i)3-s + (−0.0868 + 0.492i)4-s + (−0.462 + 0.168i)6-s + (−0.718 − 0.415i)7-s + (0.306 − 0.176i)8-s + (0.394 + 0.330i)9-s + (0.0477 + 0.0827i)11-s + (0.301 + 0.174i)12-s + (−0.00651 − 0.0178i)13-s + (0.101 + 0.578i)14-s + (−0.234 − 0.0855i)16-s + (−1.04 − 1.24i)17-s − 0.363i·18-s + (0.174 − 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.995 + 0.0981i$
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: \((2,\ 950,\ (\ :1/2),\ -0.995 + 0.0981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0385600 - 0.784084i\)
\(L(\frac12)\) \(\approx\) \(0.0385600 - 0.784084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.761 + 4.29i)T \)
good3 \( 1 + (-0.412 + 1.13i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.90 + 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.158 - 0.274i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0234 + 0.0644i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (4.30 + 5.12i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.513 + 0.0905i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.22 + 4.38i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.96 - 5.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + (2.78 + 1.01i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (5.76 - 1.01i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.59 - 6.66i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-8.96 - 1.58i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.57 - 1.32i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.313 + 1.77i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.150 - 0.179i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.314 + 1.78i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.56 + 7.05i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.32 - 1.21i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.0844 + 0.0487i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.00 + 2.18i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-4.73 - 5.64i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.508191116330852287450622875653, −9.032763969620618997715067619039, −7.898251149974086830154138421920, −7.13160833973030785041383992742, −6.65374879205494519184347205686, −5.12453109396234015504359760604, −4.06714250870872491835438963573, −2.87949454182714404023514777702, −1.92762977607468431224730221615, −0.40237402012396261038533749481, 1.74130007029500716427794930311, 3.35537087886911384375695152777, 4.17250118361235783369142481169, 5.36420777665615668800670457654, 6.30862182019634835093844343878, 6.93843045674370949524553669955, 8.172835696478615123131197912850, 8.789194657405611121733720788261, 9.637664312798531020164438236458, 10.10008333467206260767096439391

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