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Properties

Label 2-950-25.11-c1-0-5
Degree $2$
Conductor $950$
Sign $-0.968 - 0.249i$
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 + (−0.853 + 2.62i)3-s + (0.309 − 0.951i)4-s + (−0.950 − 2.02i)5-s + (−0.853 − 2.62i)6-s + 1.85·7-s + (0.309 + 0.951i)8-s + (−3.74 − 2.72i)9-s + (1.95 + 1.07i)10-s + (−1.27 + 0.928i)11-s + (2.23 + 1.62i)12-s + (4.45 + 3.23i)13-s + (−1.49 + 1.08i)14-s + (6.12 − 0.770i)15-s + (−0.809 − 0.587i)16-s + (−1.02 − 3.16i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.492 + 1.51i)3-s + (0.154 − 0.475i)4-s + (−0.425 − 0.905i)5-s + (−0.348 − 1.07i)6-s + 0.700·7-s + (0.109 + 0.336i)8-s + (−1.24 − 0.907i)9-s + (0.619 + 0.341i)10-s + (−0.385 + 0.279i)11-s + (0.645 + 0.468i)12-s + (1.23 + 0.898i)13-s + (−0.400 + 0.291i)14-s + (1.58 − 0.198i)15-s + (−0.202 − 0.146i)16-s + (−0.249 − 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.968 - 0.249i$
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: \((2,\ 950,\ (\ :1/2),\ -0.968 - 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0916875 + 0.722190i\)
\(L(\frac12)\) \(\approx\) \(0.0916875 + 0.722190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.950 + 2.02i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.853 - 2.62i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 1.85T + 7T^{2} \)
11 \( 1 + (1.27 - 0.928i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.45 - 3.23i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.02 + 3.16i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-0.609 + 0.442i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.93 - 9.03i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.70 - 8.32i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.71 + 1.97i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.353 - 0.256i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.42T + 43T^{2} \)
47 \( 1 + (0.00172 - 0.00531i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.27 - 10.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.57 + 4.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.46 - 3.24i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.181 - 0.557i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.46 + 7.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.06 - 0.773i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.43 + 7.48i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.52 + 4.68i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.45 - 3.96i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.13 - 9.64i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.53181550078776769788717764164, −9.189101248321460151959886355603, −9.118656123635454394330272252855, −8.189581129037156255981686343746, −7.13666767697176371771775559313, −5.91374666628642279320840347762, −4.97528033968631672105007292195, −4.59373351182654842327815243994, −3.48376235952020414283278095462, −1.40751703325747542626189710763, 0.47333529343675534341548942830, 1.76750571967049160205912602309, 2.79099457535892561974307636666, 4.03158711895219397181427572736, 5.76103020535016414014841673707, 6.31877046902225372366144061573, 7.31033900210482696255322951207, 8.108124413129257244147966716975, 8.245532542231520835147251916597, 9.829509651521687416200433186975

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