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Properties

Label 2-950-95.64-c1-0-6
Degree $2$
Conductor $950$
Sign $0.378 - 0.925i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 4i·7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s − 11-s + (1.73 + i)13-s + (2 + 3.46i)14-s + (−0.5 − 0.866i)16-s + (−2.59 + 1.5i)17-s + 3i·18-s + (−4 + 1.73i)19-s + (−0.866 + 0.5i)22-s + (3.46 + 2i)23-s + 1.99·26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.51i·7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s − 0.301·11-s + (0.480 + 0.277i)13-s + (0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + (−0.630 + 0.363i)17-s + 0.707i·18-s + (−0.917 + 0.397i)19-s + (−0.184 + 0.106i)22-s + (0.722 + 0.417i)23-s + 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.378 - 0.925i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48966 + 1.00068i\)
\(L(\frac12)\) \(\approx\) \(1.48966 + 1.00068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.52786402431686274141644349829, −9.164685359898488666788282213291, −8.751098376777782387180991247011, −7.73894742643701866928510943893, −6.50649615988930135671130402541, −5.65675915745214051743828982256, −5.10507947614669028854543338447, −3.89096328087104399176111917729, −2.64423249354430178508023421281, −1.94097361328050071675848570483, 0.66334955259857418275040190240, 2.60879626285126772929398887980, 3.79965129387850852672553759948, 4.37760433405627181451675830326, 5.58397859904371352651364441133, 6.55064257551126515120032214017, 7.12132474426389147860248536190, 8.077511901820438167741539878737, 8.944395515768131578852607223678, 9.930401803642043668150559966818

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