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Properties

Label 2-150-25.21-c1-0-1
Degree $2$
Conductor $150$
Sign $0.637 - 0.770i$
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  + (−0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (1.80 − 1.31i)5-s + (−0.809 + 0.587i)6-s + 2·7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (0.690 + 2.12i)10-s + (−1.61 + 4.97i)11-s + (−0.309 − 0.951i)12-s + (−1.5 − 4.61i)13-s + (−0.618 + 1.90i)14-s + 2.23·15-s + (0.309 + 0.951i)16-s + (−6.35 + 4.61i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.809 − 0.587i)5-s + (−0.330 + 0.239i)6-s + 0.755·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.218 + 0.672i)10-s + (−0.487 + 1.50i)11-s + (−0.0892 − 0.274i)12-s + (−0.416 − 1.28i)13-s + (−0.165 + 0.508i)14-s + 0.577·15-s + (0.0772 + 0.237i)16-s + (−1.54 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13165 + 0.532515i\)
\(L(\frac12)\) \(\approx\) \(1.13165 + 0.532515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (1.61 - 4.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.35 - 4.61i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.23 + 1.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.85 + 5.70i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.11 - 0.812i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3 - 2.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.663 + 2.04i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.88 + 5.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (3.85 + 2.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.92 + 5.03i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.76 - 8.50i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.85 + 5.70i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-11.4 - 8.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.972 - 2.99i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.85 - 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.427 + 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-11.2 - 8.14i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.13954061363547768375896023128, −12.61384076149909496095755422090, −10.70836891759811505260094371003, −9.991425077666941217757049930767, −8.881303827775368396817517740481, −8.095332281353903721876412967852, −6.86851847503758559561602606013, −5.28497309654397536401566236720, −4.56622908050813456726919366037, −2.13936383242117023427396859604, 1.89545622112054634888781633824, 3.16269405794208447701174193661, 4.98102964864389107454151966734, 6.52182546105273119245285082556, 7.76483694790135152754973126685, 8.967736304019538588441462549927, 9.687818995382257380037828020586, 11.28851132059886494506383363423, 11.34184022606453389640632226398, 13.10961297046229612612792468592

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