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Properties

Label 2-950-19.16-c1-0-2
Degree $2$
Conductor $950$
Sign $0.320 - 0.947i$
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  + (−0.766 − 0.642i)2-s + (−1.26 + 0.460i)3-s + (0.173 + 0.984i)4-s + (1.26 + 0.460i)6-s + (−1.43 − 2.49i)7-s + (0.500 − 0.866i)8-s + (−0.907 + 0.761i)9-s + (0.5 − 0.866i)11-s + (−0.673 − 1.16i)12-s + (−0.5 − 0.181i)13-s + (−0.500 + 2.83i)14-s + (−0.939 + 0.342i)16-s + (−1.26 − 1.06i)17-s + 1.18·18-s + (3.79 + 2.15i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.730 + 0.266i)3-s + (0.0868 + 0.492i)4-s + (0.516 + 0.188i)6-s + (−0.544 − 0.942i)7-s + (0.176 − 0.306i)8-s + (−0.302 + 0.253i)9-s + (0.150 − 0.261i)11-s + (−0.194 − 0.336i)12-s + (−0.138 − 0.0504i)13-s + (−0.133 + 0.757i)14-s + (−0.234 + 0.0855i)16-s + (−0.307 − 0.257i)17-s + 0.279·18-s + (0.869 + 0.493i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383534 + 0.275039i\)
\(L(\frac12)\) \(\approx\) \(0.383534 + 0.275039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-3.79 - 2.15i)T \)
good3 \( 1 + (1.26 - 0.460i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.181i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.26 + 1.06i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.194 - 1.10i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.81 - 1.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.847 - 1.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 + (4.85 - 1.76i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.88 - 10.6i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.39 - 1.16i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.08 - 11.8i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-6.72 - 5.64i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.08 - 6.15i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.93 - 1.62i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.02 - 5.79i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.68 - 1.70i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-8.05 + 2.93i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.32 - 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.694 + 0.252i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.88 - 7.45i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.24784152475251131372118112302, −9.646238361168700413817733043935, −8.653627037536989653065867740138, −7.69458112100005680857623700367, −6.89973771150625905089822327568, −5.92541839525655020986056996045, −4.90527476825322559656945868945, −3.83312796812927014292723683593, −2.82103807158774387208710951923, −1.11089051487234001183985372968, 0.33677539817380791883586141223, 2.06571362750334865046590796530, 3.40976056268112993770167189256, 5.02193843484980820866488481704, 5.68327622450989350869644889997, 6.52799414189203519456707359841, 7.10634077079505453559182071085, 8.317590848832165543371535202966, 9.042462440704687084603460170671, 9.704276899423786384950630778375

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