Properties

Label 2-75-25.8-c2-0-7
Degree $2$
Conductor $75$
Sign $0.694 - 0.719i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.73 + 1.39i)2-s + (1.71 + 0.270i)3-s + (3.20 + 4.40i)4-s + (−4.71 − 1.65i)5-s + (4.30 + 3.12i)6-s + (−0.250 + 0.250i)7-s + (0.696 + 4.39i)8-s + (2.85 + 0.927i)9-s + (−10.6 − 11.1i)10-s + (−2.45 − 7.55i)11-s + (4.28 + 8.40i)12-s + (−10.5 + 5.38i)13-s + (−1.03 + 0.335i)14-s + (−7.62 − 4.10i)15-s + (2.50 − 7.71i)16-s + (24.0 − 3.81i)17-s + ⋯
L(s)  = 1  + (1.36 + 0.697i)2-s + (0.570 + 0.0903i)3-s + (0.800 + 1.10i)4-s + (−0.943 − 0.330i)5-s + (0.717 + 0.521i)6-s + (−0.0357 + 0.0357i)7-s + (0.0870 + 0.549i)8-s + (0.317 + 0.103i)9-s + (−1.06 − 1.11i)10-s + (−0.223 − 0.686i)11-s + (0.356 + 0.700i)12-s + (−0.813 + 0.414i)13-s + (−0.0738 + 0.0239i)14-s + (−0.508 − 0.273i)15-s + (0.156 − 0.482i)16-s + (1.41 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.22204 + 0.943109i\)
\(L(\frac12)\) \(\approx\) \(2.22204 + 0.943109i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.270i)T \)
5 \( 1 + (4.71 + 1.65i)T \)
good2 \( 1 + (-2.73 - 1.39i)T + (2.35 + 3.23i)T^{2} \)
7 \( 1 + (0.250 - 0.250i)T - 49iT^{2} \)
11 \( 1 + (2.45 + 7.55i)T + (-97.8 + 71.1i)T^{2} \)
13 \( 1 + (10.5 - 5.38i)T + (99.3 - 136. i)T^{2} \)
17 \( 1 + (-24.0 + 3.81i)T + (274. - 89.3i)T^{2} \)
19 \( 1 + (15.4 - 21.2i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + (5.83 - 11.4i)T + (-310. - 427. i)T^{2} \)
29 \( 1 + (27.7 + 38.1i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-27.8 - 20.2i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (3.75 + 7.36i)T + (-804. + 1.10e3i)T^{2} \)
41 \( 1 + (14.5 - 44.6i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (-1.08 - 1.08i)T + 1.84e3iT^{2} \)
47 \( 1 + (-6.73 + 42.5i)T + (-2.10e3 - 682. i)T^{2} \)
53 \( 1 + (28.7 + 4.55i)T + (2.67e3 + 868. i)T^{2} \)
59 \( 1 + (-71.3 - 23.1i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-26.0 - 80.0i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + (41.9 - 6.64i)T + (4.26e3 - 1.38e3i)T^{2} \)
71 \( 1 + (-49.0 + 35.6i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-43.1 + 84.6i)T + (-3.13e3 - 4.31e3i)T^{2} \)
79 \( 1 + (62.3 + 85.8i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (15.0 + 95.2i)T + (-6.55e3 + 2.12e3i)T^{2} \)
89 \( 1 + (163. - 53.0i)T + (6.40e3 - 4.65e3i)T^{2} \)
97 \( 1 + (-16.8 + 106. i)T + (-8.94e3 - 2.90e3i)T^{2} \)
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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

−14.52971356187692605385268373442, −13.57024514711048291512854362165, −12.47617975803157539716239859500, −11.72474671185456989491777044813, −9.898538404431029247531095055611, −8.224553280462709372481201921745, −7.36732359364663086875333511207, −5.78143471942010744773942708266, −4.41507836214378098977795232144, −3.30796035544829059964084920372, 2.59234030809814376671302311752, 3.83886465010078913273021747609, 5.07858356279336419064699617630, 6.98431204824825682967045150932, 8.227947825914005976348597298720, 10.05338321865663472591091652584, 11.16466311298081835492208847345, 12.35715774125278586670135142332, 12.81336662635201284113679288581, 14.22434854882551158224427738330

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