Properties

Label 2-950-95.24-c1-0-16
Degree $2$
Conductor $950$
Sign $0.00341 - 0.999i$
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 + (3.16 + 0.558i)3-s + (−0.766 + 0.642i)4-s + (0.558 + 3.16i)6-s + (−0.0202 + 0.0116i)7-s + (−0.866 − 0.500i)8-s + (6.89 + 2.51i)9-s + (−1.08 + 1.87i)11-s + (−2.78 + 1.60i)12-s + (−1.56 + 0.276i)13-s + (−0.0179 − 0.0150i)14-s + (0.173 − 0.984i)16-s + (2.72 + 7.49i)17-s + 7.34i·18-s + (−1.06 − 4.22i)19-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (1.82 + 0.322i)3-s + (−0.383 + 0.321i)4-s + (0.227 + 1.29i)6-s + (−0.00765 + 0.00442i)7-s + (−0.306 − 0.176i)8-s + (2.29 + 0.836i)9-s + (−0.325 + 0.564i)11-s + (−0.803 + 0.464i)12-s + (−0.434 + 0.0765i)13-s + (−0.00478 − 0.00401i)14-s + (0.0434 − 0.246i)16-s + (0.661 + 1.81i)17-s + 1.73i·18-s + (−0.244 − 0.969i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00341 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00341 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.00341 - 0.999i$
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: \((2,\ 950,\ (\ :1/2),\ 0.00341 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23412 + 2.22650i\)
\(L(\frac12)\) \(\approx\) \(2.23412 + 2.22650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 + 4.22i)T \)
good3 \( 1 + (-3.16 - 0.558i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.0202 - 0.0116i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.08 - 1.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.56 - 0.276i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.72 - 7.49i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.88 + 3.43i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.09 - 2.58i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.22 + 3.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.389iT - 37T^{2} \)
41 \( 1 + (-0.972 + 5.51i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.23 + 7.43i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.46 + 4.01i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-2.43 - 2.89i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.41 - 1.24i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.84 - 7.41i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.31 + 3.62i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.85 + 8.27i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.87 - 0.330i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.65 + 9.36i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.27 + 1.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.54 + 8.74i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.81 + 10.4i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.11911366239717768429289281187, −9.054784551807519454115303731666, −8.610394424718300644671103988999, −7.75321375008727281054388395096, −7.21477680287858100176298823700, −6.03958520862971364378857218285, −4.65486513921599975277351193076, −4.04873859959990693129206459293, −2.98274428930367071583504197119, −1.98468794532812195093962807603, 1.26631898047879850727535275019, 2.57899537977672113643391422011, 3.10475364746939406380939466179, 4.09312618250791569956632667039, 5.20719847355180990028520826549, 6.57888799134927641265122393461, 7.74513992640274298805383456836, 8.074226409360314972099635940118, 9.175106640598803539224706554644, 9.676098400052523400286302211879

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