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Properties

Label 2-150-5.4-c13-0-34
Degree $2$
Conductor $150$
Sign $-0.894 + 0.447i$
Analytic cond. $160.846$
Root an. cond. $12.6825$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 64i·2-s − 729i·3-s − 4.09e3·4-s + 4.66e4·6-s − 2.63e5i·7-s − 2.62e5i·8-s − 5.31e5·9-s + 5.96e6·11-s + 2.98e6i·12-s − 3.07e7i·13-s + 1.68e7·14-s + 1.67e7·16-s − 1.42e8i·17-s − 3.40e7i·18-s − 2.89e8·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.845i·7-s − 0.353i·8-s − 0.333·9-s + 1.01·11-s + 0.288i·12-s − 1.76i·13-s + 0.597·14-s + 0.250·16-s − 1.42i·17-s − 0.235i·18-s − 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(160.846\)
Root analytic conductor: \(12.6825\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :13/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.517561650\)
\(L(\frac12)\) \(\approx\) \(1.517561650\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 64iT \)
3 \( 1 + 729iT \)
5 \( 1 \)
good7 \( 1 + 2.63e5iT - 9.68e10T^{2} \)
11 \( 1 - 5.96e6T + 3.45e13T^{2} \)
13 \( 1 + 3.07e7iT - 3.02e14T^{2} \)
17 \( 1 + 1.42e8iT - 9.90e15T^{2} \)
19 \( 1 + 2.89e8T + 4.20e16T^{2} \)
23 \( 1 + 9.85e8iT - 5.04e17T^{2} \)
29 \( 1 - 5.03e9T + 1.02e19T^{2} \)
31 \( 1 + 7.94e9T + 2.44e19T^{2} \)
37 \( 1 + 1.35e10iT - 2.43e20T^{2} \)
41 \( 1 - 1.86e10T + 9.25e20T^{2} \)
43 \( 1 - 3.33e10iT - 1.71e21T^{2} \)
47 \( 1 - 6.56e10iT - 5.46e21T^{2} \)
53 \( 1 + 2.49e11iT - 2.60e22T^{2} \)
59 \( 1 - 4.87e10T + 1.04e23T^{2} \)
61 \( 1 - 7.17e11T + 1.61e23T^{2} \)
67 \( 1 - 2.01e11iT - 5.48e23T^{2} \)
71 \( 1 - 8.72e11T + 1.16e24T^{2} \)
73 \( 1 + 1.74e12iT - 1.67e24T^{2} \)
79 \( 1 + 1.14e12T + 4.66e24T^{2} \)
83 \( 1 - 1.28e12iT - 8.87e24T^{2} \)
89 \( 1 + 6.53e12T + 2.19e25T^{2} \)
97 \( 1 - 1.16e13iT - 6.73e25T^{2} \)
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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

−10.15022964845765328646758346816, −8.886723945660477124132381214940, −7.966760507734308697685396545127, −7.03117991275684976723574970429, −6.26806494000300350396268595143, −5.02089492305847831549704528718, −3.88010913867326704877467938211, −2.56328244181267694896083834147, −0.896365769112155196958612095086, −0.35278812894125799970452229204, 1.46484143796707659905974421778, 2.23251761888813496246379419768, 3.73585943939500932467234651955, 4.32755033418447625059234556399, 5.71150374679172690303035326241, 6.73338593285059125252847134017, 8.583409572970383493347998268294, 9.019182165007068559665402515067, 10.03181943672460173220027115901, 11.13931802905050365358336667281

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