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Properties

Label 2-75-15.8-c3-0-9
Degree $2$
Conductor $75$
Sign $0.669 + 0.742i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 1.18i)2-s + (−5.11 + 0.932i)3-s − 5.17i·4-s + (−7.17 − 4.96i)6-s + (13.3 − 13.3i)7-s + (15.6 − 15.6i)8-s + (25.2 − 9.53i)9-s − 28.7i·11-s + (4.83 + 26.4i)12-s + (−14.1 − 14.1i)13-s + 31.7·14-s − 4.25·16-s + (18.5 + 18.5i)17-s + (41.3 + 18.6i)18-s − 49.0i·19-s + ⋯
L(s)  = 1  + (0.419 + 0.419i)2-s + (−0.983 + 0.179i)3-s − 0.647i·4-s + (−0.488 − 0.337i)6-s + (0.721 − 0.721i)7-s + (0.691 − 0.691i)8-s + (0.935 − 0.353i)9-s − 0.787i·11-s + (0.116 + 0.636i)12-s + (−0.302 − 0.302i)13-s + 0.605·14-s − 0.0664·16-s + (0.264 + 0.264i)17-s + (0.541 + 0.244i)18-s − 0.592i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26327 - 0.561660i\)
\(L(\frac12)\) \(\approx\) \(1.26327 - 0.561660i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.11 - 0.932i)T \)
5 \( 1 \)
good2 \( 1 + (-1.18 - 1.18i)T + 8iT^{2} \)
7 \( 1 + (-13.3 + 13.3i)T - 343iT^{2} \)
11 \( 1 + 28.7iT - 1.33e3T^{2} \)
13 \( 1 + (14.1 + 14.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-18.5 - 18.5i)T + 4.91e3iT^{2} \)
19 \( 1 + 49.0iT - 6.85e3T^{2} \)
23 \( 1 + (37.7 - 37.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 125.T + 2.43e4T^{2} \)
31 \( 1 - 247.T + 2.97e4T^{2} \)
37 \( 1 + (-127. + 127. i)T - 5.06e4iT^{2} \)
41 \( 1 - 390. iT - 6.89e4T^{2} \)
43 \( 1 + (-39.3 - 39.3i)T + 7.95e4iT^{2} \)
47 \( 1 + (-124. - 124. i)T + 1.03e5iT^{2} \)
53 \( 1 + (160. - 160. i)T - 1.48e5iT^{2} \)
59 \( 1 - 729.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (-329. + 329. i)T - 3.00e5iT^{2} \)
71 \( 1 + 171. iT - 3.57e5T^{2} \)
73 \( 1 + (-279. - 279. i)T + 3.89e5iT^{2} \)
79 \( 1 + 48.0iT - 4.93e5T^{2} \)
83 \( 1 + (-144. + 144. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + (908. - 908. i)T - 9.12e5iT^{2} \)
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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

−13.97922697436563668856020275813, −12.99018743551617826000168113890, −11.46858564047937941168916856938, −10.72255311105718620653245219052, −9.701343277851257882377132293866, −7.73330423690636504857384738042, −6.44486560968917519031586572355, −5.36383442794152895174637094167, −4.26463616948453317763866008823, −0.958252888655472075453266712888, 2.08934525185872362915185601182, 4.30545596656865106666571456559, 5.41637500864357446328764901593, 7.09342904551073333876800953753, 8.274043355187876314353592037486, 9.993063827041157420299608153749, 11.33386939690620510955120147810, 12.02373978637747555566785448863, 12.68670275420264609907367151985, 13.97119411821330236901440869674

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