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Properties

Label 2-950-95.12-c1-0-6
Degree $2$
Conductor $950$
Sign $-0.776 - 0.630i$
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  + (−0.965 − 0.258i)2-s + (−0.633 + 2.36i)3-s + (0.866 + 0.499i)4-s + (1.22 − 2.12i)6-s + (0.775 + 0.775i)7-s + (−0.707 − 0.707i)8-s + (−2.59 − 1.50i)9-s − 1.44·11-s + (−1.73 + 1.73i)12-s + (4.73 − 1.26i)13-s + (−0.548 − 0.949i)14-s + (0.500 + 0.866i)16-s + (−0.732 + 2.73i)17-s + (2.12 + 2.12i)18-s + (2.98 + 3.17i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.366 + 1.36i)3-s + (0.433 + 0.249i)4-s + (0.499 − 0.866i)6-s + (0.293 + 0.293i)7-s + (−0.249 − 0.249i)8-s + (−0.866 − 0.500i)9-s − 0.437·11-s + (−0.499 + 0.500i)12-s + (1.31 − 0.351i)13-s + (−0.146 − 0.253i)14-s + (0.125 + 0.216i)16-s + (−0.177 + 0.662i)17-s + (0.500 + 0.500i)18-s + (0.685 + 0.728i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.776 - 0.630i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.776 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297378 + 0.838290i\)
\(L(\frac12)\) \(\approx\) \(0.297378 + 0.838290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-2.98 - 3.17i)T \)
good3 \( 1 + (0.633 - 2.36i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.775 - 0.775i)T + 7iT^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 + (-4.73 + 1.26i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.732 - 2.73i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.34 - 5.01i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.85 - 6.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.778iT - 31T^{2} \)
37 \( 1 + (-3.85 + 3.85i)T - 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.1 + 2.72i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.69 + 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.20 - 1.12i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.73 - 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.81 + 2.36i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.51 - 4.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.3 - 12.3i)T - 83iT^{2} \)
89 \( 1 + (6.84 - 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 3.73i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.27881769944696931637915905293, −9.707813034335092127318491212447, −8.800157409420625689463427266443, −8.210970886471765144170504735373, −7.10017661201400455645524847501, −5.76753951458290391156119367096, −5.29426216834706218684175563057, −3.93405184555251214673456868789, −3.27570693243608140997133350644, −1.55644442533922045571199006822, 0.58055933116298930705094858691, 1.63700446068882848135528332849, 2.84928099138597458076634576877, 4.51626638712339266143952529241, 5.78696196363667019764218570771, 6.48881816916470377560779850709, 7.23825801037909677897873347154, 7.908245142644149082184730832815, 8.670018076866846742660651427796, 9.601880092093998967517682068465

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