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Properties

Label 2-75-15.14-c10-0-40
Degree $2$
Conductor $75$
Sign $0.543 + 0.839i$
Analytic cond. $47.6517$
Root an. cond. $6.90302$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26.8·2-s + (241. − 27i)3-s − 303.·4-s + (−6.48e3 + 724. i)6-s − 1.72e4i·7-s + 3.56e4·8-s + (5.75e4 − 1.30e4i)9-s + 1.86e5i·11-s + (−7.34e4 + 8.20e3i)12-s − 1.69e5i·13-s + 4.62e5i·14-s − 6.44e5·16-s + 3.43e5·17-s + (−1.54e6 + 3.49e5i)18-s + 9.49e5·19-s + ⋯
L(s)  = 1  − 0.838·2-s + (0.993 − 0.111i)3-s − 0.296·4-s + (−0.833 + 0.0931i)6-s − 1.02i·7-s + 1.08·8-s + (0.975 − 0.220i)9-s + 1.15i·11-s + (−0.295 + 0.0329i)12-s − 0.456i·13-s + 0.859i·14-s − 0.614·16-s + 0.241·17-s + (−0.817 + 0.185i)18-s + 0.383·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(47.6517\)
Root analytic conductor: \(6.90302\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :5),\ 0.543 + 0.839i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.47556 - 0.802095i\)
\(L(\frac12)\) \(\approx\) \(1.47556 - 0.802095i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-241. + 27i)T \)
5 \( 1 \)
good2 \( 1 + 26.8T + 1.02e3T^{2} \)
7 \( 1 + 1.72e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.86e5iT - 2.59e10T^{2} \)
13 \( 1 + 1.69e5iT - 1.37e11T^{2} \)
17 \( 1 - 3.43e5T + 2.01e12T^{2} \)
19 \( 1 - 9.49e5T + 6.13e12T^{2} \)
23 \( 1 - 2.65e6T + 4.14e13T^{2} \)
29 \( 1 - 3.18e6iT - 4.20e14T^{2} \)
31 \( 1 + 2.97e7T + 8.19e14T^{2} \)
37 \( 1 - 6.08e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.81e8iT - 1.34e16T^{2} \)
43 \( 1 - 1.07e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.67e8T + 5.25e16T^{2} \)
53 \( 1 - 1.92e8T + 1.74e17T^{2} \)
59 \( 1 + 6.49e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.03e9T + 7.13e17T^{2} \)
67 \( 1 + 1.87e9iT - 1.82e18T^{2} \)
71 \( 1 + 2.68e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.84e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.48e9T + 9.46e18T^{2} \)
83 \( 1 - 1.26e9T + 1.55e19T^{2} \)
89 \( 1 + 6.02e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.59e9iT - 7.37e19T^{2} \)
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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

−12.54119722280045663928285728975, −10.62281989979485763138502919181, −9.867133720260475104860677093474, −8.953362151668906578604296432838, −7.71694986154623162401623358514, −7.15609411428775993318470655971, −4.76018168905146478039834243696, −3.61522447395817311214790603496, −1.84063471934193278240357721186, −0.67060755178337734554954637385, 1.05761764242447470321250595079, 2.47853916474887472422562810483, 3.89546068302960698887427616834, 5.48073696670702644320491819588, 7.31299900644516150034138962887, 8.532519692298142264229800425002, 8.973771030003818071651105626378, 9.995859673732251454674211002565, 11.30470402494812543904828873596, 12.81013778785989896226782217678

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