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Properties

Label 2-75-15.2-c3-0-5
Degree $2$
Conductor $75$
Sign $-0.508 - 0.860i$
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  + (−3.39 + 3.39i)2-s + (5.15 + 0.624i)3-s − 15.1i·4-s + (−19.6 + 15.4i)6-s + (19.7 + 19.7i)7-s + (24.1 + 24.1i)8-s + (26.2 + 6.44i)9-s + 9.19i·11-s + (9.43 − 77.9i)12-s + (−22.4 + 22.4i)13-s − 134.·14-s − 43.3·16-s + (−50.6 + 50.6i)17-s + (−111. + 67.1i)18-s − 16.5i·19-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (0.992 + 0.120i)3-s − 1.88i·4-s + (−1.33 + 1.04i)6-s + (1.06 + 1.06i)7-s + (1.06 + 1.06i)8-s + (0.971 + 0.238i)9-s + 0.252i·11-s + (0.227 − 1.87i)12-s + (−0.478 + 0.478i)13-s − 2.55·14-s − 0.676·16-s + (−0.722 + 0.722i)17-s + (−1.45 + 0.879i)18-s − 0.200i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.508 - 0.860i$
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: \((2,\ 75,\ (\ :3/2),\ -0.508 - 0.860i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.579497 + 1.01551i\)
\(L(\frac12)\) \(\approx\) \(0.579497 + 1.01551i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.15 - 0.624i)T \)
5 \( 1 \)
good2 \( 1 + (3.39 - 3.39i)T - 8iT^{2} \)
7 \( 1 + (-19.7 - 19.7i)T + 343iT^{2} \)
11 \( 1 - 9.19iT - 1.33e3T^{2} \)
13 \( 1 + (22.4 - 22.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (50.6 - 50.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 16.5iT - 6.85e3T^{2} \)
23 \( 1 + (48.2 + 48.2i)T + 1.21e4iT^{2} \)
29 \( 1 - 203.T + 2.43e4T^{2} \)
31 \( 1 + 27.4T + 2.97e4T^{2} \)
37 \( 1 + (-130. - 130. i)T + 5.06e4iT^{2} \)
41 \( 1 + 9.19iT - 6.89e4T^{2} \)
43 \( 1 + (-63.3 + 63.3i)T - 7.95e4iT^{2} \)
47 \( 1 + (-383. + 383. i)T - 1.03e5iT^{2} \)
53 \( 1 + (441. + 441. i)T + 1.48e5iT^{2} \)
59 \( 1 - 314.T + 2.05e5T^{2} \)
61 \( 1 + 431.T + 2.26e5T^{2} \)
67 \( 1 + (649. + 649. i)T + 3.00e5iT^{2} \)
71 \( 1 + 722. iT - 3.57e5T^{2} \)
73 \( 1 + (-662. + 662. i)T - 3.89e5iT^{2} \)
79 \( 1 + 206. iT - 4.93e5T^{2} \)
83 \( 1 + (-544. - 544. i)T + 5.71e5iT^{2} \)
89 \( 1 - 563.T + 7.04e5T^{2} \)
97 \( 1 + (66.0 + 66.0i)T + 9.12e5iT^{2} \)
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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

−14.93922794598307536607910236815, −13.91632527865607546683086703491, −12.20919051913528832692255464729, −10.55480785698120118496348921750, −9.333907529969033521016945322140, −8.566761863272794012651550017565, −7.82599205979314552230287684447, −6.49746516948705248199339901889, −4.81525753196086983299697497522, −2.00864103253348985636485153044, 1.11665961754411661659409967017, 2.65591223888853710688346348200, 4.25560816627029403974895931495, 7.40385641084208135940021355476, 8.115963294431974383395927287639, 9.227942497387762172160159942276, 10.27680558271797697076035333454, 11.13654658727311485010211336086, 12.34862609864657064828699273611, 13.59153992866238066628256165882

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