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Properties

Label 2-950-95.54-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.617 + 0.786i$
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 + (0.811 + 2.22i)3-s + (−0.173 − 0.984i)4-s + (−2.22 − 0.811i)6-s + (−2.73 + 1.57i)7-s + (0.866 + 0.500i)8-s + (−2.00 + 1.68i)9-s + (−0.688 + 1.19i)11-s + (2.05 − 1.18i)12-s + (−1.47 + 4.06i)13-s + (0.547 − 3.10i)14-s + (−0.939 + 0.342i)16-s + (−0.833 + 0.993i)17-s − 2.62i·18-s + (−1.08 − 4.22i)19-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.468 + 1.28i)3-s + (−0.0868 − 0.492i)4-s + (−0.909 − 0.331i)6-s + (−1.03 + 0.596i)7-s + (0.306 + 0.176i)8-s + (−0.669 + 0.562i)9-s + (−0.207 + 0.359i)11-s + (0.592 − 0.342i)12-s + (−0.410 + 1.12i)13-s + (0.146 − 0.830i)14-s + (−0.234 + 0.0855i)16-s + (−0.202 + 0.240i)17-s − 0.618i·18-s + (−0.249 − 0.968i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.617 + 0.786i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284815 - 0.585229i\)
\(L(\frac12)\) \(\approx\) \(0.284815 - 0.585229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 4.22i)T \)
good3 \( 1 + (-0.811 - 2.22i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (2.73 - 1.57i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.688 - 1.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.47 - 4.06i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.833 - 0.993i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-2.09 + 0.369i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.0998 + 0.0837i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.173 + 0.300i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 + (10.3 - 3.76i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (11.3 + 2.00i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.0491 + 0.0585i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (6.12 - 1.08i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.27 - 5.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.36 - 7.72i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-0.662 - 0.789i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.02 + 11.4i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-4.18 - 11.5i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-13.1 + 4.79i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-9.85 + 5.68i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (17.1 + 6.23i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (10.4 - 12.4i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.21096184447906344675740782200, −9.529964842832347027365376729617, −9.111628086992010558371489506045, −8.430698051717646903455894933394, −7.08370758961683138920080065199, −6.47957115084711513910885944855, −5.25188001263714885475197908716, −4.49029061528950260112819191466, −3.42806482370389022304406691100, −2.27459993381734042948019914281, 0.32130054000815880720389780915, 1.61264538040997792851565245599, 2.89280153309157563414172359790, 3.51215788341531688788458582928, 5.14629874633427327583317832479, 6.52601021556049334287239299512, 6.97037044242431513512481983885, 8.128296458373529502706925120098, 8.283312887313968105625769224388, 9.706607881897373599064325126664

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