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Properties

Label 2-75-5.3-c8-0-7
Degree $2$
Conductor $75$
Sign $-0.973 - 0.229i$
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  + (−15.2 + 15.2i)2-s + (33.0 + 33.0i)3-s − 210. i·4-s − 1.01e3·6-s + (1.77e3 − 1.77e3i)7-s + (−688. − 688. i)8-s + 2.18e3i·9-s + 4.47e3·11-s + (6.97e3 − 6.97e3i)12-s + (−1.84e4 − 1.84e4i)13-s + 5.42e4i·14-s + 7.50e4·16-s + (−4.06e4 + 4.06e4i)17-s + (−3.34e4 − 3.34e4i)18-s + 1.61e5i·19-s + ⋯
L(s)  = 1  + (−0.954 + 0.954i)2-s + (0.408 + 0.408i)3-s − 0.823i·4-s − 0.779·6-s + (0.739 − 0.739i)7-s + (−0.168 − 0.168i)8-s + 0.333i·9-s + 0.305·11-s + (0.336 − 0.336i)12-s + (−0.644 − 0.644i)13-s + 1.41i·14-s + 1.14·16-s + (−0.486 + 0.486i)17-s + (−0.318 − 0.318i)18-s + 1.23i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.118138 + 1.01464i\)
\(L(\frac12)\) \(\approx\) \(0.118138 + 1.01464i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 - 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (15.2 - 15.2i)T - 256iT^{2} \)
7 \( 1 + (-1.77e3 + 1.77e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 4.47e3T + 2.14e8T^{2} \)
13 \( 1 + (1.84e4 + 1.84e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (4.06e4 - 4.06e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.61e5iT - 1.69e10T^{2} \)
23 \( 1 + (-2.08e5 - 2.08e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.08e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.33e6T + 8.52e11T^{2} \)
37 \( 1 + (7.78e5 - 7.78e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.79e6T + 7.98e12T^{2} \)
43 \( 1 + (3.70e6 + 3.70e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (3.40e6 - 3.40e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-3.98e6 - 3.98e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 4.53e6iT - 1.46e14T^{2} \)
61 \( 1 + 4.88e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.35e7 + 1.35e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 4.27e7T + 6.45e14T^{2} \)
73 \( 1 + (3.74e7 + 3.74e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 6.47e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.04e7 - 4.04e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 9.02e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.44e7 + 1.44e7i)T - 7.83e15iT^{2} \)
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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

−13.75389427485237135453238621324, −12.26897374066850736354581889116, −10.67791795573346427095310931373, −9.829128691215137837008650395756, −8.579896093344126633412612655954, −7.82683746248184238404311980303, −6.74715977137369432996636359883, −5.10280137786670705346776983651, −3.50042337654990188534691797954, −1.30180618150367930940723740833, 0.44474572130539680780612715354, 1.89121083728310959409013135052, 2.74398035088432893174256551724, 4.84824841732583679362487779945, 6.71926366652390511618738304514, 8.249694748196532329099022923232, 8.981510516768427350391212099168, 9.975507792888251628678708799928, 11.50388854004292408238913351110, 11.80820272403793587388964415067

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