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Properties

Label 2-75-15.14-c8-0-23
Degree $2$
Conductor $75$
Sign $0.960 - 0.278i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.03·2-s + (79.6 + 14.6i)3-s − 246.·4-s + (241. + 44.3i)6-s − 59.7i·7-s − 1.52e3·8-s + (6.13e3 + 2.32e3i)9-s − 2.18e4i·11-s + (−1.96e4 − 3.60e3i)12-s + 3.84e4i·13-s − 181. i·14-s + 5.85e4·16-s + 3.61e4·17-s + (1.86e4 + 7.06e3i)18-s + 1.36e5·19-s + ⋯
L(s)  = 1  + 0.189·2-s + (0.983 + 0.180i)3-s − 0.964·4-s + (0.186 + 0.0341i)6-s − 0.0248i·7-s − 0.372·8-s + (0.934 + 0.354i)9-s − 1.49i·11-s + (−0.948 − 0.173i)12-s + 1.34i·13-s − 0.00471i·14-s + 0.893·16-s + 0.432·17-s + (0.177 + 0.0672i)18-s + 1.04·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.960 - 0.278i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.960 - 0.278i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.59224 + 0.368438i\)
\(L(\frac12)\) \(\approx\) \(2.59224 + 0.368438i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-79.6 - 14.6i)T \)
5 \( 1 \)
good2 \( 1 - 3.03T + 256T^{2} \)
7 \( 1 + 59.7iT - 5.76e6T^{2} \)
11 \( 1 + 2.18e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.84e4iT - 8.15e8T^{2} \)
17 \( 1 - 3.61e4T + 6.97e9T^{2} \)
19 \( 1 - 1.36e5T + 1.69e10T^{2} \)
23 \( 1 - 2.35e5T + 7.83e10T^{2} \)
29 \( 1 - 8.96e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.80e4T + 8.52e11T^{2} \)
37 \( 1 + 6.65e4iT - 3.51e12T^{2} \)
41 \( 1 + 3.76e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.03e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.93e6T + 2.38e13T^{2} \)
53 \( 1 - 1.31e7T + 6.22e13T^{2} \)
59 \( 1 - 1.58e7iT - 1.46e14T^{2} \)
61 \( 1 + 3.51e6T + 1.91e14T^{2} \)
67 \( 1 - 1.39e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.51e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.93e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.15e7T + 1.51e15T^{2} \)
83 \( 1 + 4.88e7T + 2.25e15T^{2} \)
89 \( 1 + 1.09e8iT - 3.93e15T^{2} \)
97 \( 1 + 1.16e8iT - 7.83e15T^{2} \)
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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

−13.37927762400029874307816683982, −12.00146693350062166563165411912, −10.51321466131971296903701955258, −9.137100443887161469848613564836, −8.737001264791617590161519077767, −7.29378444637367135440319308877, −5.48222976976435865377239403193, −4.08687980223433922556890026480, −3.06225203280527302245897460806, −1.06909666727849983704665074238, 0.955338613602937061155409086593, 2.75999014151741419803102283195, 4.07209229463203496758608716307, 5.34338574262141228549672818593, 7.29766061087294885917959473601, 8.223727446578442371458974594195, 9.485135633233010038703472023191, 10.12779593580155943894873471915, 12.20407996726888333177198750862, 12.98392368299364781808346193771

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