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Properties

Label 2-950-95.49-c1-0-12
Degree $2$
Conductor $950$
Sign $-0.184 - 0.982i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83952 + 2.21702i\)
\(L(\frac12)\) \(\approx\) \(1.83952 + 2.21702i\)
\(L(\frac{3}{2})\) 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

−9.998682017687917682633935297317, −9.282497177561591047129457202317, −8.561072582406551863463301360369, −8.010675271842712896362115121797, −6.70045743545239157575107667944, −5.77580389717949808439564548198, −5.21153454449555762304662838103, −3.80323521017333767440659696986, −3.14228494469975446328887367978, −2.07666962745830228321772681539, 1.08445201268065939575002623277, 2.26042249001501973880793771821, 3.63136913140628961018503893458, 4.03398890125590011690737397796, 5.35872023278450675062700966495, 6.44764114606848511143064851863, 7.43929931828480698187621521405, 7.84457615306741525963217912703, 8.989566461482801073144232085273, 10.02565281908526832285503199571

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