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Properties

Label 2-75-15.2-c9-0-23
Degree $2$
Conductor $75$
Sign $-0.753 - 0.657i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−18.9 + 18.9i)2-s + (138. + 22.8i)3-s − 208i·4-s + (−3.06e3 + 2.19e3i)6-s + (6.58e3 + 6.58e3i)7-s + (−5.76e3 − 5.76e3i)8-s + (1.86e4 + 6.32e3i)9-s + 4.60e4i·11-s + (4.75e3 − 2.87e4i)12-s + (9.43e4 − 9.43e4i)13-s − 2.49e5·14-s + 3.25e5·16-s + (−1.89e5 + 1.89e5i)17-s + (−4.73e5 + 2.33e5i)18-s − 7.79e4i·19-s + ⋯
L(s)  = 1  + (−0.838 + 0.838i)2-s + (0.986 + 0.162i)3-s − 0.406i·4-s + (−0.963 + 0.690i)6-s + (1.03 + 1.03i)7-s + (−0.497 − 0.497i)8-s + (0.946 + 0.321i)9-s + 0.948i·11-s + (0.0661 − 0.400i)12-s + (0.915 − 0.915i)13-s − 1.73·14-s + 1.24·16-s + (−0.551 + 0.551i)17-s + (−1.06 + 0.524i)18-s − 0.137i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.754534 + 2.01313i\)
\(L(\frac12)\) \(\approx\) \(0.754534 + 2.01313i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-138. - 22.8i)T \)
5 \( 1 \)
good2 \( 1 + (18.9 - 18.9i)T - 512iT^{2} \)
7 \( 1 + (-6.58e3 - 6.58e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 4.60e4iT - 2.35e9T^{2} \)
13 \( 1 + (-9.43e4 + 9.43e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (1.89e5 - 1.89e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 7.79e4iT - 3.22e11T^{2} \)
23 \( 1 + (-5.11e5 - 5.11e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 4.12e6T + 1.45e13T^{2} \)
31 \( 1 - 9.87e6T + 2.64e13T^{2} \)
37 \( 1 + (4.99e6 + 4.99e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 2.87e6iT - 3.27e14T^{2} \)
43 \( 1 + (1.69e7 - 1.69e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (3.50e7 - 3.50e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-5.19e6 - 5.19e6i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.42e8T + 8.66e15T^{2} \)
61 \( 1 + 3.53e7T + 1.16e16T^{2} \)
67 \( 1 + (1.81e8 + 1.81e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 2.50e8iT - 4.58e16T^{2} \)
73 \( 1 + (-9.39e7 + 9.39e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 2.72e8iT - 1.19e17T^{2} \)
83 \( 1 + (5.21e8 + 5.21e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 6.41e8T + 3.50e17T^{2} \)
97 \( 1 + (-9.94e6 - 9.94e6i)T + 7.60e17iT^{2} \)
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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

−13.20081182858346896018497021581, −12.08651568809143161521925735559, −10.45260882029008363601148459275, −9.241203082522224363803392857574, −8.381159167945907915651330809415, −7.82168612821258193693412012093, −6.36358282408651986083678419358, −4.70027333349622252664400818313, −2.95434953994613410765785702164, −1.45141361372568404487230527010, 0.809296684304133775287152274138, 1.64428644102326329131388372475, 3.06381479489575731634150039738, 4.49401200468229990301222975007, 6.67509395206594881423554674755, 8.223057251153745749583349636683, 8.710624923628927663275542110308, 10.03472062950310291457159823310, 10.96631308203204257731321949298, 11.84767107776621908109566869446

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