Properties

Label 2-950-95.44-c1-0-28
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: $\chi_{950} (899, \cdot )$
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

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

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