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Properties

Label 2-75-5.4-c21-0-52
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93e3i·2-s + 5.90e4i·3-s − 1.65e6·4-s + 1.14e8·6-s − 4.78e7i·7-s − 8.50e8i·8-s − 3.48e9·9-s + 1.60e11·11-s − 9.79e10i·12-s − 7.86e11i·13-s − 9.26e10·14-s − 5.12e12·16-s + 2.97e12i·17-s + 6.75e12i·18-s + 2.99e13·19-s + ⋯
L(s)  = 1  − 1.33i·2-s + 0.577i·3-s − 0.790·4-s + 0.772·6-s − 0.0639i·7-s − 0.279i·8-s − 0.333·9-s + 1.86·11-s − 0.456i·12-s − 1.58i·13-s − 0.0856·14-s − 1.16·16-s + 0.357i·17-s + 0.446i·18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.720885698\)
\(L(\frac12)\) \(\approx\) \(2.720885698\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.90e4iT \)
5 \( 1 \)
good2 \( 1 + 1.93e3iT - 2.09e6T^{2} \)
7 \( 1 + 4.78e7iT - 5.58e17T^{2} \)
11 \( 1 - 1.60e11T + 7.40e21T^{2} \)
13 \( 1 + 7.86e11iT - 2.47e23T^{2} \)
17 \( 1 - 2.97e12iT - 6.90e25T^{2} \)
19 \( 1 - 2.99e13T + 7.14e26T^{2} \)
23 \( 1 + 1.91e14iT - 3.94e28T^{2} \)
29 \( 1 + 9.68e14T + 5.13e30T^{2} \)
31 \( 1 - 2.80e15T + 2.08e31T^{2} \)
37 \( 1 + 3.05e16iT - 8.55e32T^{2} \)
41 \( 1 + 2.22e16T + 7.38e33T^{2} \)
43 \( 1 - 1.63e17iT - 2.00e34T^{2} \)
47 \( 1 + 4.08e17iT - 1.30e35T^{2} \)
53 \( 1 - 4.34e17iT - 1.62e36T^{2} \)
59 \( 1 - 5.14e18T + 1.54e37T^{2} \)
61 \( 1 - 1.98e18T + 3.10e37T^{2} \)
67 \( 1 - 1.36e19iT - 2.22e38T^{2} \)
71 \( 1 - 7.35e18T + 7.52e38T^{2} \)
73 \( 1 - 6.81e19iT - 1.34e39T^{2} \)
79 \( 1 - 2.12e19T + 7.08e39T^{2} \)
83 \( 1 - 1.10e20iT - 1.99e40T^{2} \)
89 \( 1 - 7.67e18T + 8.65e40T^{2} \)
97 \( 1 + 4.63e20iT - 5.27e41T^{2} \)
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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

−10.26120026321758579593158934263, −9.541368784998478737124617765954, −8.493990978790124479460327988573, −6.91733978165854088211971508562, −5.61023175763803207998288108679, −4.20822077369177752457945192586, −3.51816363174516011300374701422, −2.56336081704307637713463716147, −1.23976188385864166976066452703, −0.56104603174521494499067560796, 1.06126423226941952388887175361, 2.01538143135935146631738753796, 3.67532065460308818396924127690, 4.88870707684333374631171521383, 6.11961512970861916435295277127, 6.79827035080618157836235437919, 7.51629634303057618781929066822, 8.828576741501014452974247345926, 9.444309217891793343083178777745, 11.56953253522063094922609009465

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