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Properties

Label 2-75-15.8-c3-0-10
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

\(L(2)\) \(\approx\) \(2.53304 + 2.00469i\)
\(L(\frac12)\) \(\approx\) \(2.53304 + 2.00469i\)
\(L(\frac{5}{2})\) 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

−14.02025439459121426460471517233, −13.62691549143496896226721932040, −12.62927993074499927163452012901, −11.60751321759295799585851908734, −9.208194089992275265727966730827, −7.977962005503125798969076929768, −7.08068238018536699206277874927, −6.02356370250665433055840566181, −4.44357443287396644337596497603, −2.90394097269245471735939559305, 2.09188174196224252437109491252, 3.55372622833743811273149660676, 4.54922500887656353602197035628, 5.97660660924973794957875499798, 8.318120844194051873027789027437, 9.948248138674941177816407463155, 10.44088047333363708529169391229, 11.77570432750707119490462763620, 12.71957474339891400649242234634, 13.90213793909309296948998031696

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