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Properties

Label 2-950-19.9-c1-0-15
Degree $2$
Conductor $950$
Sign $0.877 + 0.479i$
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  + (−0.173 − 0.984i)2-s + (0.266 − 0.223i)3-s + (−0.939 + 0.342i)4-s + (−0.266 − 0.223i)6-s + (1.87 − 3.25i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 2.83i)9-s + (2.76 + 4.79i)11-s + (−0.173 + 0.300i)12-s + (1 + 0.839i)13-s + (−3.53 − 1.28i)14-s + (0.766 − 0.642i)16-s + (0.826 + 4.68i)17-s + 2.87·18-s + (2.77 − 3.35i)19-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.153 − 0.128i)3-s + (−0.469 + 0.171i)4-s + (−0.108 − 0.0911i)6-s + (0.710 − 1.23i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.945i)9-s + (0.833 + 1.44i)11-s + (−0.0501 + 0.0868i)12-s + (0.277 + 0.232i)13-s + (−0.943 − 0.343i)14-s + (0.191 − 0.160i)16-s + (0.200 + 1.13i)17-s + 0.678·18-s + (0.637 − 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63247 - 0.417214i\)
\(L(\frac12)\) \(\approx\) \(1.63247 - 0.417214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-2.77 + 3.35i)T \)
good3 \( 1 + (-0.266 + 0.223i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.87 + 3.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.76 - 4.79i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 0.839i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.826 - 4.68i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.10 - 1.13i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.65 - 9.37i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.18 + 5.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-6.13 + 5.14i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.77 + 2.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.12 + 12.0i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.65 + 0.601i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.12 - 6.36i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.10 + 0.766i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.14 - 12.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.53 - 2.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.890 + 0.747i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-6.75 + 5.67i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.06 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.62 - 7.23i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.31 - 7.43i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.19549537716612772433562977258, −9.259638276017157327547981409573, −8.311242674686101931022122387538, −7.50267793814403176188433543496, −6.89005709507359017256264934873, −5.34471413597201985331606563130, −4.39852369050662808573588602638, −3.77767195414585000705498801551, −2.15575826254469816567246124244, −1.32126804562551901740252325131, 0.995160498437921818062091050918, 2.80765206523762642311214092362, 3.83886718575228161587351234862, 5.07475371948277599806634076077, 6.02865494643431868119590751162, 6.34941727072058121284540309170, 7.924423309584590797115059125231, 8.297437931683477541538187809112, 9.290664667031731743860109600066, 9.589445552072771641562275902822

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