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Properties

Label 2-75-15.2-c3-0-6
Degree $2$
Conductor $75$
Sign $0.563 - 0.826i$
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  + (1.39 − 1.39i)2-s + (1.62 + 4.93i)3-s + 4.10i·4-s + (9.15 + 4.61i)6-s + (−3.80 − 3.80i)7-s + (16.8 + 16.8i)8-s + (−21.7 + 16.0i)9-s + 61.8i·11-s + (−20.2 + 6.67i)12-s + (48.1 − 48.1i)13-s − 10.6·14-s + 14.3·16-s + (47.5 − 47.5i)17-s + (−7.89 + 52.7i)18-s − 93.4i·19-s + ⋯
L(s)  = 1  + (0.493 − 0.493i)2-s + (0.312 + 0.949i)3-s + 0.513i·4-s + (0.623 + 0.314i)6-s + (−0.205 − 0.205i)7-s + (0.746 + 0.746i)8-s + (−0.804 + 0.594i)9-s + 1.69i·11-s + (−0.487 + 0.160i)12-s + (1.02 − 1.02i)13-s − 0.202·14-s + 0.223·16-s + (0.679 − 0.679i)17-s + (−0.103 + 0.690i)18-s − 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.563 - 0.826i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.80275 + 0.953008i\)
\(L(\frac12)\) \(\approx\) \(1.80275 + 0.953008i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 4.93i)T \)
5 \( 1 \)
good2 \( 1 + (-1.39 + 1.39i)T - 8iT^{2} \)
7 \( 1 + (3.80 + 3.80i)T + 343iT^{2} \)
11 \( 1 - 61.8iT - 1.33e3T^{2} \)
13 \( 1 + (-48.1 + 48.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-47.5 + 47.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 93.4iT - 6.85e3T^{2} \)
23 \( 1 + (33.7 + 33.7i)T + 1.21e4iT^{2} \)
29 \( 1 - 179.T + 2.43e4T^{2} \)
31 \( 1 + 123.T + 2.97e4T^{2} \)
37 \( 1 + (10.5 + 10.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 61.8iT - 6.89e4T^{2} \)
43 \( 1 + (-133. + 133. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-56.9 + 56.9i)T - 1.03e5iT^{2} \)
53 \( 1 + (-234. - 234. i)T + 1.48e5iT^{2} \)
59 \( 1 + 260.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (320. + 320. i)T + 3.00e5iT^{2} \)
71 \( 1 - 1.08e3iT - 3.57e5T^{2} \)
73 \( 1 + (208. - 208. i)T - 3.89e5iT^{2} \)
79 \( 1 - 676. iT - 4.93e5T^{2} \)
83 \( 1 + (-71.3 - 71.3i)T + 5.71e5iT^{2} \)
89 \( 1 - 228.T + 7.04e5T^{2} \)
97 \( 1 + (89.5 + 89.5i)T + 9.12e5iT^{2} \)
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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

−14.09461059604336492173246864374, −13.11027638691587769582317023882, −12.08350066568355351022998532077, −10.86830191121354799721108228455, −9.933696613995021834326845455147, −8.597616151670953633574132618433, −7.33111712624435139621099887792, −5.14868879341993690914294025136, −4.01534681628962122412798593819, −2.70531315402687563174459278236, 1.30539047205114265947596520820, 3.61889870192235946146850859774, 5.85876521521359169020844906897, 6.34505455574176825829102302108, 7.936483738621796113787136953469, 9.046361863635731024994861923656, 10.67243182320430952582559110172, 11.87698481100169139609807773468, 13.17268022969306725956848602130, 13.96623277918303160663669332869

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