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Properties

Label 2-150-25.19-c1-0-2
Degree $2$
Conductor $150$
Sign $0.974 - 0.225i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
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  + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.166 + 2.22i)5-s + (0.809 + 0.587i)6-s − 2.07i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.530 + 2.17i)10-s + (0.160 + 0.494i)11-s + (0.951 + 0.309i)12-s + (−2.07 − 0.675i)13-s + (−0.642 − 1.97i)14-s + (−1.90 + 1.17i)15-s + (0.309 − 0.951i)16-s + (1.58 − 2.18i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.0746 + 0.997i)5-s + (0.330 + 0.239i)6-s − 0.785i·7-s + (0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.167 + 0.686i)10-s + (0.0484 + 0.149i)11-s + (0.274 + 0.0892i)12-s + (−0.576 − 0.187i)13-s + (−0.171 − 0.528i)14-s + (−0.491 + 0.303i)15-s + (0.0772 − 0.237i)16-s + (0.384 − 0.528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.974 - 0.225i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66549 + 0.190273i\)
\(L(\frac12)\) \(\approx\) \(1.66549 + 0.190273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (0.166 - 2.22i)T \)
good7 \( 1 + 2.07iT - 7T^{2} \)
11 \( 1 + (-0.160 - 0.494i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.07 + 0.675i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.58 + 2.18i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.55 + 4.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.67 + 1.19i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (7.30 - 5.30i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.99 - 4.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (5.04 + 1.64i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.996 - 3.06i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.53iT - 43T^{2} \)
47 \( 1 + (-5.44 - 7.49i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.43 + 1.97i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.67 - 8.22i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.88 - 11.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.93 + 5.41i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-6.60 + 4.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.65 + 1.18i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.84 + 2.06i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.71 + 10.6i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.04 - 9.37i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.32 - 8.70i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.25736202780856522039435762894, −12.05564862990500772959903738071, −10.78223322511534004117700036607, −10.45713055026890340304086348532, −9.122621741359314426490211681377, −7.48322894167670073512067482602, −6.69896461041228368041805692190, −5.05356691651534727532203987559, −3.82274750758155501537675217923, −2.62218181372433609251246945968, 2.10304965717315225729640861085, 3.92387088891517629959245578434, 5.30440925796893097663807519466, 6.31676008283695478996744564947, 7.82134132181401199735867757555, 8.593086669269589357102929260979, 9.736228551487315412734762513803, 11.40370012959080150144268339738, 12.36992955906251677802269684007, 12.83262699557625380437605109669

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