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Properties

Label 2-950-19.7-c1-0-17
Degree $2$
Conductor $950$
Sign $0.658 + 0.752i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 4·7-s + 0.999·8-s + (1.5 − 2.59i)9-s − 11-s + (−1 + 1.73i)13-s + (−2 − 3.46i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s − 3·18-s + (4 − 1.73i)19-s + (0.5 + 0.866i)22-s + (−2 + 3.46i)23-s + 1.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.51·7-s + 0.353·8-s + (0.5 − 0.866i)9-s − 0.301·11-s + (−0.277 + 0.480i)13-s + (−0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s − 0.707·18-s + (0.917 − 0.397i)19-s + (0.106 + 0.184i)22-s + (−0.417 + 0.722i)23-s + 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44240 - 0.654035i\)
\(L(\frac12)\) \(\approx\) \(1.44240 - 0.654035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-4 + 1.73i)T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.806945868444657179890078391459, −9.331278319024421238205381058932, −8.160092209846945261948423177638, −7.74449496139154088767916504511, −6.64712956023782217981691171034, −5.37910592099521989355386862677, −4.48585311121576836448253852871, −3.55849083310874862777676318559, −2.13962698116482825132393866715, −1.09921874520690091350756836785, 1.21933920243924547826322587540, 2.47815361197550296732342454157, 4.21259459093084646413143670001, 5.13721948123108281567837603195, 5.58134872982081990061053675284, 7.16683350197101172488231284634, 7.59955754260840875186267303582, 8.304064873109822080349928709952, 9.131648018292945956548558624444, 10.38450045593392763058529638412

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