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Properties

Label 2-950-25.11-c1-0-17
Degree $2$
Conductor $950$
Sign $0.857 - 0.514i$
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 + (−0.457 + 1.40i)3-s + (0.309 − 0.951i)4-s + (1.65 + 1.49i)5-s + (0.457 + 1.40i)6-s + 0.448·7-s + (−0.309 − 0.951i)8-s + (0.656 + 0.477i)9-s + (2.22 + 0.238i)10-s + (2.18 − 1.58i)11-s + (1.19 + 0.869i)12-s + (0.731 + 0.531i)13-s + (0.363 − 0.263i)14-s + (−2.86 + 1.64i)15-s + (−0.809 − 0.587i)16-s + (−0.924 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.263 + 0.812i)3-s + (0.154 − 0.475i)4-s + (0.741 + 0.670i)5-s + (0.186 + 0.574i)6-s + 0.169·7-s + (−0.109 − 0.336i)8-s + (0.218 + 0.159i)9-s + (0.703 + 0.0753i)10-s + (0.658 − 0.478i)11-s + (0.345 + 0.251i)12-s + (0.202 + 0.147i)13-s + (0.0970 − 0.0705i)14-s + (−0.740 + 0.425i)15-s + (−0.202 − 0.146i)16-s + (−0.224 − 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.857 - 0.514i$
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: \((2,\ 950,\ (\ :1/2),\ 0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38676 + 0.661089i\)
\(L(\frac12)\) \(\approx\) \(2.38676 + 0.661089i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 1.49i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.457 - 1.40i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.448T + 7T^{2} \)
11 \( 1 + (-2.18 + 1.58i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.731 - 0.531i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.924 + 2.84i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-3.84 + 2.79i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.40 - 7.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.996 - 3.06i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.61 + 1.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.00 - 2.18i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + (-3.04 + 9.36i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.33 - 4.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.561 + 0.407i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.326 - 0.237i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.03 + 3.17i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.48 - 4.58i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.79 + 2.02i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.51 - 4.65i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.24 + 3.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.17 + 3.75i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.242 - 0.747i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.33169518737892093657865919803, −9.465401822490538079689143105120, −8.842252217781689787926136034262, −7.28532083947525717265521887572, −6.54823700212758487396763939768, −5.55137475635486320196713280954, −4.84492283818776982206209615092, −3.80569871900859466266917487153, −2.87403242131890919766781681950, −1.54213553309848814322249634181, 1.18467595062677489698962282700, 2.24894355843614400041609453887, 3.89585381335940337583726566453, 4.76558565933956373566754232367, 5.88206125350065884778411497608, 6.34969875538896606942921901027, 7.29970423403878360956566851209, 8.073805469400346570624136775254, 9.140129123344469173498291674942, 9.754380474863415643478180927460

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