Properties

Label 2-75-5.4-c1-0-0
Degree $2$
Conductor $75$
Sign $-0.447 - 0.894i$
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  + 2i·2-s + i·3-s − 2·4-s − 2·6-s − 3i·7-s − 9-s + 2·11-s − 2i·12-s i·13-s + 6·14-s − 4·16-s + 2i·17-s − 2i·18-s + 5·19-s + 3·21-s + 4i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s + 0.577i·3-s − 4-s − 0.816·6-s − 1.13i·7-s − 0.333·9-s + 0.603·11-s − 0.577i·12-s − 0.277i·13-s + 1.60·14-s − 16-s + 0.485i·17-s − 0.471i·18-s + 1.14·19-s + 0.654·21-s + 0.852i·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 75,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490983 + 0.794428i\)
\(L(\frac12)\) \(\approx\) \(0.490983 + 0.794428i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 17iT - 97T^{2} \)
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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

−14.91080771710218934571399627491, −14.26732083822925441082468851828, −13.21515765633412552018180043720, −11.47012906506001366604380880864, −10.28050345487040540000848756328, −8.998851506099230354236081802260, −7.75232414789649101971239372127, −6.73196839614468195283312946179, −5.36869590813685455208628735030, −3.95467184940752398421650199006, 1.87406895516215122089339214148, 3.42992584500941121446531627913, 5.50602029648065362711011946692, 7.19132413134035066790980104844, 8.942075476039868462130469896912, 9.715877541712923973372115428251, 11.44560299852911851411715448857, 11.74278915826813398513090580853, 12.82684476259162027079655811408, 13.76560097149990409305510756326

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