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Properties

Label 2-75-15.8-c3-0-4
Degree $2$
Conductor $75$
Sign $-0.229 + 0.973i$
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  + (−3.67 − 3.67i)2-s + (−3.67 + 3.67i)3-s + 19i·4-s + 27·6-s + (40.4 − 40.4i)8-s − 27i·9-s + (−69.8 − 69.8i)12-s − 145.·16-s + (14.6 + 14.6i)17-s + (−99.2 + 99.2i)18-s − 164i·19-s + (139. − 139. i)23-s + 297. i·24-s + (99.2 + 99.2i)27-s + 232·31-s + (209. + 209. i)32-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + (−0.707 + 0.707i)3-s + 2.37i·4-s + 1.83·6-s + (1.78 − 1.78i)8-s − i·9-s + (−1.67 − 1.67i)12-s − 2.26·16-s + (0.209 + 0.209i)17-s + (−1.29 + 1.29i)18-s − 1.98i·19-s + (1.26 − 1.26i)23-s + 2.52i·24-s + (0.707 + 0.707i)27-s + 1.34·31-s + (1.15 + 1.15i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(4.42514\)
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: \((2,\ 75,\ (\ :3/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.312411 - 0.394749i\)
\(L(\frac12)\) \(\approx\) \(0.312411 - 0.394749i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.67 - 3.67i)T \)
5 \( 1 \)
good2 \( 1 + (3.67 + 3.67i)T + 8iT^{2} \)
7 \( 1 - 343iT^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + 2.19e3iT^{2} \)
17 \( 1 + (-14.6 - 14.6i)T + 4.91e3iT^{2} \)
19 \( 1 + 164iT - 6.85e3T^{2} \)
23 \( 1 + (-139. + 139. i)T - 1.21e4iT^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 232T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4iT^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 7.95e4iT^{2} \)
47 \( 1 + (242. + 242. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-323. + 323. i)T - 1.48e5iT^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 358T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5iT^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + 3.89e5iT^{2} \)
79 \( 1 + 304iT - 4.93e5T^{2} \)
83 \( 1 + (-580. + 580. i)T - 5.71e5iT^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5iT^{2} \)
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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

−13.17845273392295125693971935032, −12.10481325145126433027667246258, −11.19006334447788947892940279763, −10.49012998412797378468241327541, −9.437699951302257258045107546301, −8.563402332066144946439928253125, −6.84702350477602627744975740930, −4.62142724150123569012342760278, −2.92458588307240902277531611796, −0.63488197754918008055775487603, 1.26109311214373982530178842783, 5.31562020905777169364986863503, 6.31323880811286934801440676876, 7.42097399454339030479681129817, 8.254581961082140733500939770161, 9.678463731623077614938127577386, 10.70149638314292583124999886630, 12.00063921132314638248351644628, 13.53050336215637811872143977892, 14.64952742764110574050599897698

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