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Properties

Label 2-75-3.2-c8-0-6
Degree $2$
Conductor $75$
Sign $0.555 - 0.831i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.4i·2-s + (−45 + 67.3i)3-s − 248·4-s + (1.51e3 + 1.01e3i)6-s + 1.75e3·7-s − 179. i·8-s + (−2.51e3 − 6.06e3i)9-s + 6.95e3i·11-s + (1.11e4 − 1.67e4i)12-s − 2.57e4·13-s − 3.92e4i·14-s − 6.75e4·16-s − 7.48e4i·17-s + (−1.36e5 + 5.63e4i)18-s + 1.89e4·19-s + ⋯
L(s)  = 1  − 1.40i·2-s + (−0.555 + 0.831i)3-s − 0.968·4-s + (1.16 + 0.779i)6-s + 0.728·7-s − 0.0438i·8-s + (−0.382 − 0.923i)9-s + 0.475i·11-s + (0.538 − 0.805i)12-s − 0.900·13-s − 1.02i·14-s − 1.03·16-s − 0.896i·17-s + (−1.29 + 0.536i)18-s + 0.145·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.621602 + 0.332260i\)
\(L(\frac12)\) \(\approx\) \(0.621602 + 0.332260i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (45 - 67.3i)T \)
5 \( 1 \)
good2 \( 1 + 22.4iT - 256T^{2} \)
7 \( 1 - 1.75e3T + 5.76e6T^{2} \)
11 \( 1 - 6.95e3iT - 2.14e8T^{2} \)
13 \( 1 + 2.57e4T + 8.15e8T^{2} \)
17 \( 1 + 7.48e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.89e4T + 1.69e10T^{2} \)
23 \( 1 - 4.70e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.60e5iT - 5.00e11T^{2} \)
31 \( 1 + 3.51e5T + 8.52e11T^{2} \)
37 \( 1 + 1.33e6T + 3.51e12T^{2} \)
41 \( 1 - 1.87e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.52e6T + 1.16e13T^{2} \)
47 \( 1 - 4.08e6iT - 2.38e13T^{2} \)
53 \( 1 - 6.60e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.37e7iT - 1.46e14T^{2} \)
61 \( 1 - 7.53e5T + 1.91e14T^{2} \)
67 \( 1 + 2.26e6T + 4.06e14T^{2} \)
71 \( 1 - 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.76e7T + 8.06e14T^{2} \)
79 \( 1 + 2.29e7T + 1.51e15T^{2} \)
83 \( 1 - 4.63e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.26e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.47e8T + 7.83e15T^{2} \)
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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

−12.52339831250955989265689714946, −11.69757800541747014829048492897, −11.00236211332466004255926181015, −9.917739891777601320788017082330, −9.199949216526917596029386024782, −7.26397622977707208402442583836, −5.29539665397744324323830716469, −4.27872858250621541514994820717, −2.89974313146140004823428344933, −1.33365843388245525136650483603, 0.24164688123621780406836533867, 2.10203239877848330549034157547, 4.73450197968475733688084905638, 5.78749089268642832044809250103, 6.80378412140734767112081217749, 7.82391932586227961826660662715, 8.624654265334267569770350615686, 10.60147295225520905143866702319, 11.73431469155489004038660274857, 12.87612685381098833680385724101

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