Properties

Label 2-75-5.2-c4-0-9
Degree $2$
Conductor $75$
Sign $-0.326 + 0.945i$
Analytic cond. $7.75274$
Root an. cond. $2.78437$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.550 + 0.550i)2-s + (3.67 − 3.67i)3-s − 15.3i·4-s + 4.04·6-s + (−16.7 − 16.7i)7-s + (17.2 − 17.2i)8-s − 27i·9-s − 197.·11-s + (−56.5 − 56.5i)12-s + (120. − 120. i)13-s − 18.4i·14-s − 227.·16-s + (152. + 152. i)17-s + (14.8 − 14.8i)18-s − 418. i·19-s + ⋯
L(s)  = 1  + (0.137 + 0.137i)2-s + (0.408 − 0.408i)3-s − 0.962i·4-s + 0.112·6-s + (−0.342 − 0.342i)7-s + (0.270 − 0.270i)8-s − 0.333i·9-s − 1.62·11-s + (−0.392 − 0.392i)12-s + (0.713 − 0.713i)13-s − 0.0942i·14-s − 0.887·16-s + (0.527 + 0.527i)17-s + (0.0458 − 0.0458i)18-s − 1.15i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(7.75274\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :2),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.952068 - 1.33600i\)
\(L(\frac12)\) \(\approx\) \(0.952068 - 1.33600i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.67 + 3.67i)T \)
5 \( 1 \)
good2 \( 1 + (-0.550 - 0.550i)T + 16iT^{2} \)
7 \( 1 + (16.7 + 16.7i)T + 2.40e3iT^{2} \)
11 \( 1 + 197.T + 1.46e4T^{2} \)
13 \( 1 + (-120. + 120. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-152. - 152. i)T + 8.35e4iT^{2} \)
19 \( 1 + 418. iT - 1.30e5T^{2} \)
23 \( 1 + (-621. + 621. i)T - 2.79e5iT^{2} \)
29 \( 1 - 792. iT - 7.07e5T^{2} \)
31 \( 1 - 208.T + 9.23e5T^{2} \)
37 \( 1 + (-460. - 460. i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.43e3T + 2.82e6T^{2} \)
43 \( 1 + (2.11e3 - 2.11e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-2.91e3 - 2.91e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-1.28e3 + 1.28e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.95e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.22e3T + 1.38e7T^{2} \)
67 \( 1 + (1.16e3 + 1.16e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.10e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.45e3 + 3.45e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 9.87e3iT - 3.89e7T^{2} \)
83 \( 1 + (-3.29e3 + 3.29e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 6.13e3iT - 6.27e7T^{2} \)
97 \( 1 + (728. + 728. i)T + 8.85e7iT^{2} \)
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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

−13.30860198864017051984095877075, −12.91265812886890509694817190243, −10.93042991587268866387836998929, −10.27048321671059571923990712761, −8.843724750716145123372309248533, −7.52112756615786694659073238391, −6.23471920923343731872231333052, −4.93165534704508484177756569623, −2.82645911041639594851907513604, −0.78400697358742381036765637018, 2.58415679100320949687313944764, 3.82347679930203734266924845493, 5.44804549377077555154835647813, 7.38683416606481138186595484350, 8.358933109208481745966260820498, 9.538737429040725809379306574300, 10.85993530016528662826108740997, 12.03014752201309898084128637266, 13.13471384643832229347760660545, 13.85720994544317089772363467647

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