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Properties

Label 2-75-5.4-c17-0-12
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 500. i·2-s + 6.56e3i·3-s − 1.19e5·4-s − 3.28e6·6-s − 8.90e6i·7-s + 5.87e6i·8-s − 4.30e7·9-s + 4.48e8·11-s − 7.82e8i·12-s − 3.89e9i·13-s + 4.45e9·14-s − 1.85e10·16-s + 5.96e8i·17-s − 2.15e10i·18-s + 4.36e10·19-s + ⋯
L(s)  = 1  + 1.38i·2-s + 0.577i·3-s − 0.910·4-s − 0.798·6-s − 0.583i·7-s + 0.123i·8-s − 0.333·9-s + 0.631·11-s − 0.525i·12-s − 1.32i·13-s + 0.807·14-s − 1.08·16-s + 0.0207i·17-s − 0.460i·18-s + 0.590·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.651578179\)
\(L(\frac12)\) \(\approx\) \(1.651578179\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 6.56e3iT \)
5 \( 1 \)
good2 \( 1 - 500. iT - 1.31e5T^{2} \)
7 \( 1 + 8.90e6iT - 2.32e14T^{2} \)
11 \( 1 - 4.48e8T + 5.05e17T^{2} \)
13 \( 1 + 3.89e9iT - 8.65e18T^{2} \)
17 \( 1 - 5.96e8iT - 8.27e20T^{2} \)
19 \( 1 - 4.36e10T + 5.48e21T^{2} \)
23 \( 1 - 7.23e10iT - 1.41e23T^{2} \)
29 \( 1 + 1.82e12T + 7.25e24T^{2} \)
31 \( 1 - 5.27e12T + 2.25e25T^{2} \)
37 \( 1 - 1.62e13iT - 4.56e26T^{2} \)
41 \( 1 - 9.78e12T + 2.61e27T^{2} \)
43 \( 1 - 1.46e14iT - 5.87e27T^{2} \)
47 \( 1 + 2.43e14iT - 2.66e28T^{2} \)
53 \( 1 - 6.84e14iT - 2.05e29T^{2} \)
59 \( 1 + 9.83e14T + 1.27e30T^{2} \)
61 \( 1 + 1.84e15T + 2.24e30T^{2} \)
67 \( 1 - 2.46e15iT - 1.10e31T^{2} \)
71 \( 1 - 9.43e14T + 2.96e31T^{2} \)
73 \( 1 - 1.15e16iT - 4.74e31T^{2} \)
79 \( 1 + 1.58e16T + 1.81e32T^{2} \)
83 \( 1 - 1.62e16iT - 4.21e32T^{2} \)
89 \( 1 - 4.03e16T + 1.37e33T^{2} \)
97 \( 1 - 7.85e16iT - 5.95e33T^{2} \)
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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

−11.72849812654162323205337029855, −10.55674277681754344291334151989, −9.448061227569374164022927108728, −8.264439103731031703105466396058, −7.42063204924507520042316814068, −6.25942954912743136488254218490, −5.31378524632494569743909085458, −4.24270725711736619058191562741, −2.92943511503006041710005270472, −1.01718394903131148071660950046, 0.36448101604039340291851399190, 1.52166807932030689912084403833, 2.21455340608177936535070440047, 3.37057697364086095170876987251, 4.52431489574096805464574809499, 6.12972119986952247852951777550, 7.22410813899399574108840961332, 8.840180541015521384962355449835, 9.542508842136686017838405649550, 10.89012702485756998412961816210

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