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Properties

Label 2-75-15.2-c9-0-2
Degree $2$
Conductor $75$
Sign $-0.170 + 0.985i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.02 + 4.02i)2-s + (−130. + 52.2i)3-s + 479. i·4-s + (313. − 734. i)6-s + (−17.2 − 17.2i)7-s + (−3.99e3 − 3.99e3i)8-s + (1.42e4 − 1.36e4i)9-s + 8.84e4i·11-s + (−2.50e4 − 6.24e4i)12-s + (−1.06e5 + 1.06e5i)13-s + 139.·14-s − 2.13e5·16-s + (−2.21e5 + 2.21e5i)17-s + (−2.40e3 + 1.12e5i)18-s + 1.38e5i·19-s + ⋯
L(s)  = 1  + (−0.177 + 0.177i)2-s + (−0.927 + 0.372i)3-s + 0.936i·4-s + (0.0988 − 0.231i)6-s + (−0.00272 − 0.00272i)7-s + (−0.344 − 0.344i)8-s + (0.722 − 0.691i)9-s + 1.82i·11-s + (−0.349 − 0.869i)12-s + (−1.03 + 1.03i)13-s + 0.000968·14-s − 0.813·16-s + (−0.641 + 0.641i)17-s + (−0.00540 + 0.251i)18-s + 0.243i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.170 + 0.985i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.170 + 0.985i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.288630 - 0.342960i\)
\(L(\frac12)\) \(\approx\) \(0.288630 - 0.342960i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (130. - 52.2i)T \)
5 \( 1 \)
good2 \( 1 + (4.02 - 4.02i)T - 512iT^{2} \)
7 \( 1 + (17.2 + 17.2i)T + 4.03e7iT^{2} \)
11 \( 1 - 8.84e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.06e5 - 1.06e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (2.21e5 - 2.21e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 1.38e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.10e6 - 1.10e6i)T + 1.80e12iT^{2} \)
29 \( 1 + 2.89e6T + 1.45e13T^{2} \)
31 \( 1 - 3.99e6T + 2.64e13T^{2} \)
37 \( 1 + (4.00e6 + 4.00e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 9.96e6iT - 3.27e14T^{2} \)
43 \( 1 + (1.56e6 - 1.56e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.81e7 - 1.81e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-1.37e7 - 1.37e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 7.44e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (7.61e7 + 7.61e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.68e7iT - 4.58e16T^{2} \)
73 \( 1 + (-1.01e8 + 1.01e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.01e8iT - 1.19e17T^{2} \)
83 \( 1 + (1.97e7 + 1.97e7i)T + 1.86e17iT^{2} \)
89 \( 1 + 7.06e8T + 3.50e17T^{2} \)
97 \( 1 + (-1.62e8 - 1.62e8i)T + 7.60e17iT^{2} \)
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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

−13.04264195519244979872740914220, −12.25543722748813239879638462023, −11.47791743571217806379246285292, −10.00980072760465652753149643310, −9.142534794686362783584525515374, −7.40111152780783479866779204979, −6.72685241974224749653045059103, −4.90617011375170839915169786852, −3.97894558737613576172538158698, −1.99776476967888005704471713680, 0.19876904471168878881409703467, 0.903113531931144562360131682867, 2.65173106782010561736332018951, 4.92450517743602698328925992560, 5.78437712323228417526550865403, 6.89198089711297903147655659467, 8.469855233161106017093396082284, 9.902419182177915591286411151669, 10.90540226251002750143611725551, 11.53804414602996216401321304182

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