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Properties

Label 2-950-95.64-c1-0-20
Degree $2$
Conductor $950$
Sign $0.722 + 0.691i$
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 + (2.21 − 1.28i)3-s + (0.499 − 0.866i)4-s + (−1.28 + 2.21i)6-s + 0.438i·7-s + 0.999i·8-s + (1.78 − 3.08i)9-s + 11-s − 2.56i·12-s + (−1.73 − i)13-s + (−0.219 − 0.379i)14-s + (−0.5 − 0.866i)16-s + (4.43 − 2.56i)17-s + 3.56i·18-s + (2.5 − 3.57i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (1.28 − 0.739i)3-s + (0.249 − 0.433i)4-s + (−0.522 + 0.905i)6-s + 0.165i·7-s + 0.353i·8-s + (0.593 − 1.02i)9-s + 0.301·11-s − 0.739i·12-s + (−0.480 − 0.277i)13-s + (−0.0585 − 0.101i)14-s + (−0.125 − 0.216i)16-s + (1.07 − 0.621i)17-s + 0.839i·18-s + (0.573 − 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72414 - 0.692369i\)
\(L(\frac12)\) \(\approx\) \(1.72414 - 0.692369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 1.28i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.438iT - 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.43 + 2.56i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.05 + 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 4.68iT - 37T^{2} \)
41 \( 1 + (-3.06 - 5.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.70 - 1.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.49 + 1.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.54 + 3.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.28 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 - 4.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.17 + 4.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.12 + 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.45 + 0.842i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.56 + 9.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (1.34 - 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.45 + 0.842i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.646926069411934797325866826755, −9.007945666900873800403786964431, −8.136405303068910010960278471686, −7.64680988913941479146962518066, −6.88066820660953467435701615788, −5.88988901216426542823839998536, −4.64210100487068099805976303921, −3.13327750597882642498346383346, −2.38838964476666400701231461527, −1.02112526799805556563942386418, 1.53222304177150500109006465160, 2.80649472652805392429985314933, 3.61374450524243981506358456389, 4.44026095649428932292389337134, 5.83678762032070239435363628428, 7.10901770018863945036409642801, 8.084472904778275295620736220913, 8.405947084806833085338599311457, 9.556577250390664456691835917942, 9.872919131747413244541194206291

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