Properties

Label 2-950-5.4-c1-0-16
Degree $2$
Conductor $950$
Sign $-0.447 + 0.894i$
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  i·2-s − 1.77i·3-s − 4-s − 1.77·6-s + 2.69i·7-s + i·8-s − 0.144·9-s + 5.54·11-s + 1.77i·12-s − 2.91i·13-s + 2.69·14-s + 16-s − 4.91i·17-s + 0.144i·18-s − 19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.02i·3-s − 0.5·4-s − 0.723·6-s + 1.01i·7-s + 0.353i·8-s − 0.0483·9-s + 1.67·11-s + 0.511i·12-s − 0.809i·13-s + 0.719·14-s + 0.250·16-s − 1.19i·17-s + 0.0341i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871625 - 1.41032i\)
\(L(\frac12)\) \(\approx\) \(0.871625 - 1.41032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 1.77iT - 3T^{2} \)
7 \( 1 - 2.69iT - 7T^{2} \)
11 \( 1 - 5.54T + 11T^{2} \)
13 \( 1 + 2.91iT - 13T^{2} \)
17 \( 1 + 4.91iT - 17T^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 + 1.08T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 + 4.54iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 9.54iT - 43T^{2} \)
47 \( 1 - 0.836iT - 47T^{2} \)
53 \( 1 - 9.78iT - 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 + 2.85iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 5.15iT - 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 1.71iT - 83T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−9.568467128120567785415241232297, −9.117880400138790770687371730668, −8.158017599197913552061478965094, −7.24851569276400909066275386894, −6.35748611916909983362952119576, −5.49294426597085922542007122156, −4.28615626997913101189517988385, −3.06556509149052843203318753050, −2.03683891189127788840186396488, −0.936750481410795245148169208529, 1.37384753281162657188321274580, 3.57170476355457033795059513807, 4.23274293104942551334939936796, 4.73437425656294287503114842348, 6.36635678655076905848975237776, 6.61413045910446523553128376259, 7.81269380801745631115106418347, 8.752874293113246485128328649730, 9.468598072593009167919550115519, 10.14478038990527302647938908680

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