Properties

Label 2-150-75.23-c1-0-8
Degree $2$
Conductor $150$
Sign $0.911 + 0.410i$
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  + (0.987 + 0.156i)2-s + (−0.428 − 1.67i)3-s + (0.951 + 0.309i)4-s + (0.334 + 2.21i)5-s + (−0.161 − 1.72i)6-s + (2.78 − 2.78i)7-s + (0.891 + 0.453i)8-s + (−2.63 + 1.43i)9-s + (−0.0154 + 2.23i)10-s + (−2.08 − 2.86i)11-s + (0.110 − 1.72i)12-s + (0.331 + 2.09i)13-s + (3.18 − 2.31i)14-s + (3.56 − 1.50i)15-s + (0.809 + 0.587i)16-s + (−0.476 + 0.934i)17-s + ⋯
L(s)  = 1  + (0.698 + 0.110i)2-s + (−0.247 − 0.968i)3-s + (0.475 + 0.154i)4-s + (0.149 + 0.988i)5-s + (−0.0657 − 0.704i)6-s + (1.05 − 1.05i)7-s + (0.315 + 0.160i)8-s + (−0.877 + 0.479i)9-s + (−0.00488 + 0.707i)10-s + (−0.627 − 0.863i)11-s + (0.0319 − 0.498i)12-s + (0.0920 + 0.581i)13-s + (0.850 − 0.617i)14-s + (0.920 − 0.389i)15-s + (0.202 + 0.146i)16-s + (−0.115 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53691 - 0.330259i\)
\(L(\frac12)\) \(\approx\) \(1.53691 - 0.330259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
3 \( 1 + (0.428 + 1.67i)T \)
5 \( 1 + (-0.334 - 2.21i)T \)
good7 \( 1 + (-2.78 + 2.78i)T - 7iT^{2} \)
11 \( 1 + (2.08 + 2.86i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.331 - 2.09i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.476 - 0.934i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (4.71 - 1.53i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.22 - 7.70i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-0.858 + 2.64i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.458 - 1.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.07 - 0.169i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-0.691 + 0.952i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (8.25 + 8.25i)T + 43iT^{2} \)
47 \( 1 + (-10.2 + 5.24i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (1.93 + 3.79i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (5.11 + 3.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.09 + 0.794i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-8.29 - 4.22i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-2.47 - 0.805i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.82 - 1.39i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (14.4 + 4.69i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-9.56 - 4.87i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-2.14 + 1.55i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-7.23 - 14.2i)T + (-57.0 + 78.4i)T^{2} \)
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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

−13.32687149657997147687800217931, −11.86386867983139638711159233065, −11.11555585319103123348133689102, −10.46110640455093846558476562688, −8.282576462877742756500873438569, −7.44341946969268474312801776152, −6.53053811183917990322217212945, −5.40703615867717999497166627292, −3.73034065464028500348342913637, −1.99183010242691724516390697663, 2.41705907191402501038989933587, 4.52233759740745688375140405146, 4.98591538380810201285799836876, 6.07279363030910183413647413191, 8.121716466843875303505809544864, 8.953696410879916077817936647168, 10.21211970499525502471533925055, 11.16676924880918660779675146638, 12.23358206769151258310412530141, 12.79556004974366794506803178752

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