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Properties

Label 2-75-5.4-c17-0-34
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 670. i·2-s − 6.56e3i·3-s − 3.18e5·4-s + 4.39e6·6-s + 2.70e7i·7-s − 1.25e8i·8-s − 4.30e7·9-s + 1.01e9·11-s + 2.09e9i·12-s − 1.04e9i·13-s − 1.81e10·14-s + 4.25e10·16-s − 2.00e10i·17-s − 2.88e10i·18-s − 1.64e10·19-s + ⋯
L(s)  = 1  + 1.85i·2-s − 0.577i·3-s − 2.43·4-s + 1.06·6-s + 1.77i·7-s − 2.65i·8-s − 0.333·9-s + 1.42·11-s + 1.40i·12-s − 0.356i·13-s − 3.28·14-s + 2.47·16-s − 0.695i·17-s − 0.617i·18-s − 0.222·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.8464354716\)
\(L(\frac12)\) \(\approx\) \(0.8464354716\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 6.56e3iT \)
5 \( 1 \)
good2 \( 1 - 670. iT - 1.31e5T^{2} \)
7 \( 1 - 2.70e7iT - 2.32e14T^{2} \)
11 \( 1 - 1.01e9T + 5.05e17T^{2} \)
13 \( 1 + 1.04e9iT - 8.65e18T^{2} \)
17 \( 1 + 2.00e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.64e10T + 5.48e21T^{2} \)
23 \( 1 - 5.92e11iT - 1.41e23T^{2} \)
29 \( 1 + 4.83e11T + 7.25e24T^{2} \)
31 \( 1 + 5.20e12T + 2.25e25T^{2} \)
37 \( 1 + 3.82e13iT - 4.56e26T^{2} \)
41 \( 1 + 6.56e13T + 2.61e27T^{2} \)
43 \( 1 + 5.83e13iT - 5.87e27T^{2} \)
47 \( 1 + 7.43e12iT - 2.66e28T^{2} \)
53 \( 1 + 3.01e14iT - 2.05e29T^{2} \)
59 \( 1 - 3.44e14T + 1.27e30T^{2} \)
61 \( 1 + 2.43e15T + 2.24e30T^{2} \)
67 \( 1 + 1.19e15iT - 1.10e31T^{2} \)
71 \( 1 - 5.42e15T + 2.96e31T^{2} \)
73 \( 1 + 5.84e15iT - 4.74e31T^{2} \)
79 \( 1 + 2.03e15T + 1.81e32T^{2} \)
83 \( 1 + 1.18e16iT - 4.21e32T^{2} \)
89 \( 1 - 2.43e16T + 1.37e33T^{2} \)
97 \( 1 - 1.06e17iT - 5.95e33T^{2} \)
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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

−11.65370032847245446869005377095, −9.269107798520050556372146689987, −8.986951519832751400439525155282, −7.78086712602719985354918549276, −6.78430647952541454918627684399, −5.83618451595910146860688654436, −5.20927350530011611641417250358, −3.56853914523764386412216337877, −1.80706405796626756300877193467, −0.20868788133135048411350741991, 0.902730941519237865559893619735, 1.68736402943775546471103914485, 3.26010324502087607586258006061, 4.06478888634505481129645612234, 4.59115642816428693679541230907, 6.63180365032451352367206597920, 8.387974134280039249100518106084, 9.463824690202077841972183817260, 10.33421120434751738659039399931, 10.95184350496223902099335303278

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