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Properties

Label 2-150-5.4-c7-0-11
Degree $2$
Conductor $150$
Sign $0.894 + 0.447i$
Analytic cond. $46.8577$
Root an. cond. $6.84527$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s + 27i·3-s − 64·4-s + 216·6-s + 391i·7-s + 512i·8-s − 729·9-s − 4.39e3·11-s − 1.72e3i·12-s − 1.34e4i·13-s + 3.12e3·14-s + 4.09e3·16-s + 7.68e3i·17-s + 5.83e3i·18-s + 1.37e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.430i·7-s + 0.353i·8-s − 0.333·9-s − 0.996·11-s − 0.288i·12-s − 1.69i·13-s + 0.304·14-s + 0.250·16-s + 0.379i·17-s + 0.235i·18-s + 0.458·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(46.8577\)
Root analytic conductor: \(6.84527\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :7/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.618834736\)
\(L(\frac12)\) \(\approx\) \(1.618834736\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
3 \( 1 - 27iT \)
5 \( 1 \)
good7 \( 1 - 391iT - 8.23e5T^{2} \)
11 \( 1 + 4.39e3T + 1.94e7T^{2} \)
13 \( 1 + 1.34e4iT - 6.27e7T^{2} \)
17 \( 1 - 7.68e3iT - 4.10e8T^{2} \)
19 \( 1 - 1.37e4T + 8.93e8T^{2} \)
23 \( 1 - 3.54e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.57e5T + 1.72e10T^{2} \)
31 \( 1 + 9.93e4T + 2.75e10T^{2} \)
37 \( 1 - 1.61e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.21e5T + 1.94e11T^{2} \)
43 \( 1 - 3.40e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.08e4iT - 5.06e11T^{2} \)
53 \( 1 + 8.91e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.34e6T + 2.48e12T^{2} \)
61 \( 1 - 3.39e6T + 3.14e12T^{2} \)
67 \( 1 - 2.24e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.73e6T + 9.09e12T^{2} \)
73 \( 1 + 5.02e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.57e6T + 1.92e13T^{2} \)
83 \( 1 + 7.79e6iT - 2.71e13T^{2} \)
89 \( 1 - 5.80e6T + 4.42e13T^{2} \)
97 \( 1 - 2.49e6iT - 8.07e13T^{2} \)
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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

−11.48519671570809561493621341679, −10.49337007390083128974833537127, −9.896978957823366808642658464928, −8.624979881973633056590674474981, −7.73475764715182246712930746004, −5.77619748568146310290896091864, −4.97291923013368513672036060904, −3.45130182480188653390703064844, −2.50810715866138551598039632187, −0.69533343277918214669614747826, 0.73982100135571957176577749057, 2.37418218062608443963522803983, 4.12644510167032123690826295023, 5.34760769385700164135427999730, 6.65218933767494223036784120148, 7.36339014793102259309513929543, 8.453387167078567504802141829034, 9.504950163020247138574651077647, 10.75026838569408361694920302226, 11.90677760786218491724298961118

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