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Properties

Label 2-950-95.24-c1-0-5
Degree $2$
Conductor $950$
Sign $0.941 + 0.336i$
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  + (−0.342 − 0.939i)2-s + (−1.85 − 0.326i)3-s + (−0.766 + 0.642i)4-s + (0.326 + 1.85i)6-s + (−2.65 + 1.53i)7-s + (0.866 + 0.500i)8-s + (0.5 + 0.181i)9-s + (2.17 − 3.76i)11-s + (1.62 − 0.939i)12-s + (−5.67 + i)13-s + (2.34 + 1.96i)14-s + (0.173 − 0.984i)16-s + (−0.705 − 1.93i)17-s − 0.532i·18-s + (−4.34 − 0.405i)19-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (−1.06 − 0.188i)3-s + (−0.383 + 0.321i)4-s + (0.133 + 0.755i)6-s + (−1.00 + 0.579i)7-s + (0.306 + 0.176i)8-s + (0.166 + 0.0606i)9-s + (0.655 − 1.13i)11-s + (0.469 − 0.271i)12-s + (−1.57 + 0.277i)13-s + (0.627 + 0.526i)14-s + (0.0434 − 0.246i)16-s + (−0.171 − 0.470i)17-s − 0.125i·18-s + (−0.995 − 0.0929i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541011 - 0.0937709i\)
\(L(\frac12)\) \(\approx\) \(0.541011 - 0.0937709i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (4.34 + 0.405i)T \)
good3 \( 1 + (1.85 + 0.326i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (2.65 - 1.53i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.67 - i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.705 + 1.93i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-4.98 - 5.94i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.87 - 1.41i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.22 - 7.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (0.0248 - 0.140i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.99 + 8.34i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.01 + 5.53i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.25 - 3.87i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-7.19 + 2.61i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (6.94 - 5.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.15 + 5.92i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.34 - 3.64i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.07 + 0.365i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.0641 + 0.363i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.652 + 0.376i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.308 + 1.75i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-3.65 - 10.0i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.15024816284682731298639088179, −9.117700132607252365818211626358, −8.785296152611945936216186137187, −7.23784597254247101446255389305, −6.55156608632297108738912170378, −5.64623213583314435063340687347, −4.82338006792951156612405635777, −3.43745460249120346777691906311, −2.50576784411544634074070045644, −0.72919425099920595018028078923, 0.51919524426680490637262646038, 2.57649801935901285190809377758, 4.36768958701052574975791649002, 4.73214029617625270178859529582, 6.05918459990072334688999329997, 6.59608154398219978627057122542, 7.26175819270105615208070704650, 8.335953687427637787642685276529, 9.497633621613033416033268137917, 10.03674552403315541297233507373

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