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Properties

Label 2-950-25.16-c1-0-11
Degree $2$
Conductor $950$
Sign $-0.669 - 0.743i$
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 + (0.862 + 2.65i)3-s + (0.309 + 0.951i)4-s + (1.23 + 1.86i)5-s + (0.862 − 2.65i)6-s + 2.25·7-s + (0.309 − 0.951i)8-s + (−3.87 + 2.81i)9-s + (0.0938 − 2.23i)10-s + (0.0254 + 0.0185i)11-s + (−2.25 + 1.64i)12-s + (−4.67 + 3.39i)13-s + (−1.82 − 1.32i)14-s + (−3.87 + 4.89i)15-s + (−0.809 + 0.587i)16-s + (0.0990 − 0.304i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.498 + 1.53i)3-s + (0.154 + 0.475i)4-s + (0.553 + 0.832i)5-s + (0.352 − 1.08i)6-s + 0.850·7-s + (0.109 − 0.336i)8-s + (−1.29 + 0.938i)9-s + (0.0296 − 0.706i)10-s + (0.00768 + 0.00558i)11-s + (−0.651 + 0.473i)12-s + (−1.29 + 0.942i)13-s + (−0.486 − 0.353i)14-s + (−1.00 + 1.26i)15-s + (−0.202 + 0.146i)16-s + (0.0240 − 0.0739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.669 - 0.743i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601658 + 1.35103i\)
\(L(\frac12)\) \(\approx\) \(0.601658 + 1.35103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.862 - 2.65i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.25T + 7T^{2} \)
11 \( 1 + (-0.0254 - 0.0185i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.67 - 3.39i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.0990 + 0.304i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (-0.0332 - 0.0241i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.178 - 0.550i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.68 + 8.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.24 + 1.63i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.27 + 4.56i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.572T + 43T^{2} \)
47 \( 1 + (-3.36 - 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.213 - 0.655i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.25 - 5.99i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.76 - 2.73i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.82 + 11.7i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.00173 + 0.00535i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.15 - 4.47i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.63 + 11.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.30 - 7.10i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.25 + 2.36i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.85 + 5.71i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.18502732087919286057638415477, −9.546911529520890525237272988956, −9.142248456904711562845826858008, −8.003675576605281431583929450647, −7.25188606067936661862140585255, −5.94718779487597101541639900186, −4.76054494314945551057578097127, −4.08489107646146670501197817936, −2.86308470405314917604607558200, −2.07842684399620033359977135518, 0.78350601638639696552727966630, 1.79024494016556401175949305914, 2.71687819034594913423407174636, 4.77693768596823404618196637280, 5.53171186142402405466245918849, 6.57633340909267030330423656735, 7.38134739067454566615995032586, 8.130077309796710757721806275450, 8.508073158608928828797766476241, 9.507269922358771993156415515866

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