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Properties

Label 2-75-5.4-c1-0-2
Degree $2$
Conductor $75$
Sign $0.894 + 0.447i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s + 4-s + 6-s − 3i·8-s − 9-s − 4·11-s + i·12-s + 2i·13-s − 16-s + 2i·17-s + i·18-s − 4·19-s + 4i·22-s + 3·24-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + 0.408·6-s − 1.06i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 0.554i·13-s − 0.250·16-s + 0.485i·17-s + 0.235i·18-s − 0.917·19-s + 0.852i·22-s + 0.612·24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982545 - 0.231947i\)
\(L(\frac12)\) \(\approx\) \(0.982545 - 0.231947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
good2 \( 1 + iT - 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−14.56809004589719542184363179136, −13.12990429596880409303772465604, −12.19251527247341498052026940081, −10.91103772036436981090062822670, −10.40433512253293882895387196372, −9.066268324289895262971809662894, −7.52156633457050854866829125129, −5.95632579760025348206124952315, −4.17314498147057539715128367537, −2.50577656933738056614132106187, 2.59227805750375734226524276590, 5.21819757671916085151264765739, 6.43506067565300041466428256615, 7.59119348029160234777087323207, 8.436983745411476596171759227745, 10.30336279752583334125713240413, 11.37619323081471910438366922608, 12.59166450815821389290892738729, 13.60501366116669137684821098129, 14.82600110360146444091838197656

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