Properties

Label 2-75-15.14-c2-0-4
Degree $2$
Conductor $75$
Sign $-0.121 + 0.992i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.31·2-s + (1.65 − 2.5i)3-s + 7·4-s + (−5.5 + 8.29i)6-s − 9.94·8-s + (−3.5 − 8.29i)9-s − 16.5i·11-s + (11.6 − 17.5i)12-s − 10i·13-s + 5.00·16-s − 3.31·17-s + (11.6 + 27.4i)18-s − 7·19-s + 55.0i·22-s + 19.8·23-s + (−16.5 + 24.8i)24-s + ⋯
L(s)  = 1  − 1.65·2-s + (0.552 − 0.833i)3-s + 1.75·4-s + (−0.916 + 1.38i)6-s − 1.24·8-s + (−0.388 − 0.921i)9-s − 1.50i·11-s + (0.967 − 1.45i)12-s − 0.769i·13-s + 0.312·16-s − 0.195·17-s + (0.644 + 1.52i)18-s − 0.368·19-s + 2.50i·22-s + 0.865·23-s + (−0.687 + 1.03i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ -0.121 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.423227 - 0.478306i\)
\(L(\frac12)\) \(\approx\) \(0.423227 - 0.478306i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 2.5i)T \)
5 \( 1 \)
good2 \( 1 + 3.31T + 4T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 + 16.5iT - 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 + 3.31T + 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 - 19.8T + 529T^{2} \)
29 \( 1 - 33.1iT - 841T^{2} \)
31 \( 1 - 42T + 961T^{2} \)
37 \( 1 + 40iT - 1.36e3T^{2} \)
41 \( 1 - 16.5iT - 1.68e3T^{2} \)
43 \( 1 - 50iT - 1.84e3T^{2} \)
47 \( 1 - 46.4T + 2.20e3T^{2} \)
53 \( 1 + 46.4T + 2.80e3T^{2} \)
59 \( 1 - 66.3iT - 3.48e3T^{2} \)
61 \( 1 + 8T + 3.72e3T^{2} \)
67 \( 1 - 45iT - 4.48e3T^{2} \)
71 \( 1 + 33.1iT - 5.04e3T^{2} \)
73 \( 1 - 35iT - 5.32e3T^{2} \)
79 \( 1 + 12T + 6.24e3T^{2} \)
83 \( 1 - 69.6T + 6.88e3T^{2} \)
89 \( 1 + 149. iT - 7.92e3T^{2} \)
97 \( 1 + 70iT - 9.40e3T^{2} \)
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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

−13.98736952966368462425964918412, −12.85905828912534486385178688375, −11.46011071517585769267112723919, −10.50981825181986785872294517575, −9.050575231139934831003180986404, −8.414257734253821439877396532851, −7.39831720023179465024042714305, −6.14761488436847336722925105830, −2.88055083721362980639495000710, −0.908229494399085101493476265249, 2.22481081588170554204089271884, 4.51991831460946341657674476768, 6.85375806299344915528016160779, 8.028262890006306496922678387450, 9.117818490591680144961085547601, 9.840655984491762970981446057829, 10.72465697462911655549271894807, 11.95389046428366435455292877285, 13.69968483671011841339864106806, 15.13288639524818421349808910885

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