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Properties

Label 2-75-5.2-c8-0-18
Degree $2$
Conductor $75$
Sign $0.229 + 0.973i$
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  + (11.1 + 11.1i)2-s + (−33.0 + 33.0i)3-s − 9.11i·4-s − 734.·6-s + (1.15e3 + 1.15e3i)7-s + (2.94e3 − 2.94e3i)8-s − 2.18e3i·9-s − 2.49e4·11-s + (301. + 301. i)12-s + (−1.53e4 + 1.53e4i)13-s + 2.57e4i·14-s + 6.31e4·16-s + (−6.85e4 − 6.85e4i)17-s + (2.42e4 − 2.42e4i)18-s − 1.59e5i·19-s + ⋯
L(s)  = 1  + (0.694 + 0.694i)2-s + (−0.408 + 0.408i)3-s − 0.0356i·4-s − 0.566·6-s + (0.482 + 0.482i)7-s + (0.719 − 0.719i)8-s − 0.333i·9-s − 1.70·11-s + (0.0145 + 0.0145i)12-s + (−0.538 + 0.538i)13-s + 0.670i·14-s + 0.963·16-s + (−0.821 − 0.821i)17-s + (0.231 − 0.231i)18-s − 1.22i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.882344 - 0.698303i\)
\(L(\frac12)\) \(\approx\) \(0.882344 - 0.698303i\)
\(L(5)\) 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 \( 1 + (-11.1 - 11.1i)T + 256iT^{2} \)
7 \( 1 + (-1.15e3 - 1.15e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 2.49e4T + 2.14e8T^{2} \)
13 \( 1 + (1.53e4 - 1.53e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (6.85e4 + 6.85e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 1.59e5iT - 1.69e10T^{2} \)
23 \( 1 + (-1.12e5 + 1.12e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 8.24e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.43e5T + 8.52e11T^{2} \)
37 \( 1 + (5.64e5 + 5.64e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 3.69e6T + 7.98e12T^{2} \)
43 \( 1 + (2.72e6 - 2.72e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (2.57e6 + 2.57e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (9.93e5 - 9.93e5i)T - 6.22e13iT^{2} \)
59 \( 1 + 8.43e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.18e7T + 1.91e14T^{2} \)
67 \( 1 + (1.90e7 + 1.90e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 6.06e6T + 6.45e14T^{2} \)
73 \( 1 + (7.87e6 - 7.87e6i)T - 8.06e14iT^{2} \)
79 \( 1 - 2.50e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.76e7 + 4.76e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 6.58e7iT - 3.93e15T^{2} \)
97 \( 1 + (2.70e7 + 2.70e7i)T + 7.83e15iT^{2} \)
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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

−12.94154669935300394173296438401, −11.51301495364516562007707343762, −10.51714857413904835515062572172, −9.299205706804613917763000507174, −7.69842949624207142249570157819, −6.47905546928272282519587230149, −5.13399786699740738579089259310, −4.65690229426474766957918956442, −2.48319774821309078934305300285, −0.27597395575569880256402242445, 1.63811634375229586405612110912, 2.96250902828486770695774078335, 4.53067659403222242712354633502, 5.56954217798901949779679687233, 7.44239515670595962682955662454, 8.213306720567706076225969643678, 10.42421615291971652250575144592, 10.92675320413075321523697510326, 12.25784723227043124677812277807, 12.93668523801689988938761549449

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