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Properties

Label 2-950-95.74-c1-0-21
Degree $2$
Conductor $950$
Sign $0.0367 + 0.999i$
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  + (−0.984 − 0.173i)2-s + (0.984 − 1.17i)3-s + (0.939 + 0.342i)4-s + (−1.17 + 0.984i)6-s + (1.62 − 0.939i)7-s + (−0.866 − 0.5i)8-s + (0.113 + 0.642i)9-s + (0.0812 − 0.140i)11-s + (1.32 − 0.766i)12-s + (−3.03 − 3.61i)13-s + (−1.76 + 0.642i)14-s + (0.766 + 0.642i)16-s + (4.28 + 0.754i)17-s − 0.652i·18-s + (−2.77 − 3.35i)19-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.568 − 0.677i)3-s + (0.469 + 0.171i)4-s + (−0.479 + 0.402i)6-s + (0.615 − 0.355i)7-s + (−0.306 − 0.176i)8-s + (0.0377 + 0.214i)9-s + (0.0244 − 0.0424i)11-s + (0.383 − 0.221i)12-s + (−0.840 − 1.00i)13-s + (−0.471 + 0.171i)14-s + (0.191 + 0.160i)16-s + (1.03 + 0.183i)17-s − 0.153i·18-s + (−0.637 − 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.0367 + 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.0367 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01695 - 0.980221i\)
\(L(\frac12)\) \(\approx\) \(1.01695 - 0.980221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (2.77 + 3.35i)T \)
good3 \( 1 + (-0.984 + 1.17i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.62 + 0.939i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0812 + 0.140i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.03 + 3.61i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-4.28 - 0.754i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.08 + 5.73i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.414 + 2.35i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.81 - 3.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.36iT - 37T^{2} \)
41 \( 1 + (-1.81 - 1.52i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.223 + 0.613i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.77 + 1.19i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.03 + 5.58i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.708 + 4.01i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.21 + 0.805i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.61 + 0.284i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.39 + 0.872i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (10.0 - 12.0i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.885 - 0.511i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.57 + 2.15i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (17.6 + 3.10i)T + (91.1 + 33.1i)T^{2} \)
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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

−9.892238351592762719147223635235, −8.753308762360166396790292665428, −8.153317802887933863834750626535, −7.50620541696687026511782764789, −6.85633896813906603853889416928, −5.54974343762722152683082751330, −4.49453611444633066163735866840, −2.98896345567208615340042053534, −2.15956039327903730270226181175, −0.825970938700416656955628979906, 1.51447910091000410072548422733, 2.77678894817117658443190662671, 3.92171311249112491790496525094, 4.94262025853112422000441146817, 5.99487433769361078994433729198, 7.09875811253256737123280813642, 7.896457496751354012024473513519, 8.725669470132824908069857369370, 9.432804668716331024839921111398, 9.938100369978480851806793471101

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