Properties

Label 2-150-1.1-c5-0-8
Degree $2$
Conductor $150$
Sign $1$
Analytic cond. $24.0575$
Root an. cond. $4.90485$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s + 36·6-s + 47·7-s + 64·8-s + 81·9-s + 222·11-s + 144·12-s + 101·13-s + 188·14-s + 256·16-s + 162·17-s + 324·18-s + 1.68e3·19-s + 423·21-s + 888·22-s + 306·23-s + 576·24-s + 404·26-s + 729·27-s + 752·28-s + 7.89e3·29-s − 8.59e3·31-s + 1.02e3·32-s + 1.99e3·33-s + 648·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.362·7-s + 0.353·8-s + 1/3·9-s + 0.553·11-s + 0.288·12-s + 0.165·13-s + 0.256·14-s + 1/4·16-s + 0.135·17-s + 0.235·18-s + 1.07·19-s + 0.209·21-s + 0.391·22-s + 0.120·23-s + 0.204·24-s + 0.117·26-s + 0.192·27-s + 0.181·28-s + 1.74·29-s − 1.60·31-s + 0.176·32-s + 0.319·33-s + 0.0961·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(24.0575\)
Root analytic conductor: \(4.90485\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.105786123\)
\(L(\frac12)\) \(\approx\) \(4.105786123\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 - p^{2} T \)
5 \( 1 \)
good7 \( 1 - 47 T + p^{5} T^{2} \)
11 \( 1 - 222 T + p^{5} T^{2} \)
13 \( 1 - 101 T + p^{5} T^{2} \)
17 \( 1 - 162 T + p^{5} T^{2} \)
19 \( 1 - 1685 T + p^{5} T^{2} \)
23 \( 1 - 306 T + p^{5} T^{2} \)
29 \( 1 - 7890 T + p^{5} T^{2} \)
31 \( 1 + 8593 T + p^{5} T^{2} \)
37 \( 1 - 8642 T + p^{5} T^{2} \)
41 \( 1 + 18168 T + p^{5} T^{2} \)
43 \( 1 - 14351 T + p^{5} T^{2} \)
47 \( 1 + 1098 T + p^{5} T^{2} \)
53 \( 1 - 17916 T + p^{5} T^{2} \)
59 \( 1 - 17610 T + p^{5} T^{2} \)
61 \( 1 + 21853 T + p^{5} T^{2} \)
67 \( 1 - 107 T + p^{5} T^{2} \)
71 \( 1 + 40728 T + p^{5} T^{2} \)
73 \( 1 - 34706 T + p^{5} T^{2} \)
79 \( 1 + 69160 T + p^{5} T^{2} \)
83 \( 1 + 108534 T + p^{5} T^{2} \)
89 \( 1 - 35040 T + p^{5} T^{2} \)
97 \( 1 + 823 T + p^{5} T^{2} \)
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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

−12.18452460335810328998248419726, −11.31526889831787521891073353130, −10.10827384837243017461556577652, −8.952107049992930790764298024546, −7.80029746826116766041276396731, −6.72677953396481098455839922497, −5.36500444508342913110942955917, −4.11484627014223843786486899812, −2.91215246599475002606626735673, −1.37257291352487699055648867815, 1.37257291352487699055648867815, 2.91215246599475002606626735673, 4.11484627014223843786486899812, 5.36500444508342913110942955917, 6.72677953396481098455839922497, 7.80029746826116766041276396731, 8.952107049992930790764298024546, 10.10827384837243017461556577652, 11.31526889831787521891073353130, 12.18452460335810328998248419726

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