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Properties

Label 2-950-25.11-c1-0-14
Degree $2$
Conductor $950$
Sign $-0.434 - 0.900i$
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  + (−0.809 + 0.587i)2-s + (−0.924 + 2.84i)3-s + (0.309 − 0.951i)4-s + (2.23 − 0.0110i)5-s + (−0.924 − 2.84i)6-s + 2.08·7-s + (0.309 + 0.951i)8-s + (−4.81 − 3.49i)9-s + (−1.80 + 1.32i)10-s + (3.40 − 2.47i)11-s + (2.42 + 1.75i)12-s + (−0.805 − 0.585i)13-s + (−1.68 + 1.22i)14-s + (−2.03 + 6.37i)15-s + (−0.809 − 0.587i)16-s + (1.62 + 4.99i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.533 + 1.64i)3-s + (0.154 − 0.475i)4-s + (0.999 − 0.00495i)5-s + (−0.377 − 1.16i)6-s + 0.788·7-s + (0.109 + 0.336i)8-s + (−1.60 − 1.16i)9-s + (−0.569 + 0.418i)10-s + (1.02 − 0.744i)11-s + (0.698 + 0.507i)12-s + (−0.223 − 0.162i)13-s + (−0.451 + 0.327i)14-s + (−0.525 + 1.64i)15-s + (−0.202 − 0.146i)16-s + (0.393 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723935 + 1.15332i\)
\(L(\frac12)\) \(\approx\) \(0.723935 + 1.15332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-2.23 + 0.0110i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.924 - 2.84i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 2.08T + 7T^{2} \)
11 \( 1 + (-3.40 + 2.47i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.805 + 0.585i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.62 - 4.99i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-2.99 + 2.17i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.246 + 0.759i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.42 - 7.46i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.69 - 5.58i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.680 - 0.494i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.529T + 43T^{2} \)
47 \( 1 + (-1.35 + 4.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.04 - 6.30i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (11.0 + 8.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.874 + 0.635i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.37 + 7.30i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.68 - 5.19i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.09 - 5.15i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.07 + 6.37i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.75 - 11.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.61 - 1.90i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.0530 + 0.163i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.26728644442640412033315112563, −9.497490029754767796852590968277, −8.865506451116678382485757783360, −8.171664031915059812055886759723, −6.57038708740270577312940368083, −5.97028217002208764638949637398, −5.12148101424363946455219483167, −4.38276012541457281116586009487, −3.09533944097027746027680941248, −1.31603919278927903713702518604, 1.00802517033471778355203377203, 1.79325492718382810295682948030, 2.66408167515787204818917451837, 4.60302200286826411866317338501, 5.70117843687592818058926613090, 6.52184161916154652671762905127, 7.33356586600950213973321566282, 7.84543297353837923817144575380, 9.084898646443395189901569139907, 9.604632117718119849777326776640

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