Properties

Label 2-75-15.14-c2-0-7
Degree $2$
Conductor $75$
Sign $0.929 + 0.368i$
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  + 2.23·2-s + (2.23 − 2i)3-s + 1.00·4-s + (5.00 − 4.47i)6-s + 6i·7-s − 6.70·8-s + (1.00 − 8.94i)9-s − 4.47i·11-s + (2.23 − 2.00i)12-s + 16i·13-s + 13.4i·14-s − 19·16-s − 4.47·17-s + (2.23 − 20.0i)18-s + 2·19-s + ⋯
L(s)  = 1  + 1.11·2-s + (0.745 − 0.666i)3-s + 0.250·4-s + (0.833 − 0.745i)6-s + 0.857i·7-s − 0.838·8-s + (0.111 − 0.993i)9-s − 0.406i·11-s + (0.186 − 0.166i)12-s + 1.23i·13-s + 0.958i·14-s − 1.18·16-s − 0.263·17-s + (0.124 − 1.11i)18-s + 0.105·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.22757 - 0.425429i\)
\(L(\frac12)\) \(\approx\) \(2.22757 - 0.425429i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.23 + 2i)T \)
5 \( 1 \)
good2 \( 1 - 2.23T + 4T^{2} \)
7 \( 1 - 6iT - 49T^{2} \)
11 \( 1 + 4.47iT - 121T^{2} \)
13 \( 1 - 16iT - 169T^{2} \)
17 \( 1 + 4.47T + 289T^{2} \)
19 \( 1 - 2T + 361T^{2} \)
23 \( 1 - 13.4T + 529T^{2} \)
29 \( 1 + 31.3iT - 841T^{2} \)
31 \( 1 + 18T + 961T^{2} \)
37 \( 1 - 16iT - 1.36e3T^{2} \)
41 \( 1 + 62.6iT - 1.68e3T^{2} \)
43 \( 1 - 16iT - 1.84e3T^{2} \)
47 \( 1 - 49.1T + 2.20e3T^{2} \)
53 \( 1 - 4.47T + 2.80e3T^{2} \)
59 \( 1 - 4.47iT - 3.48e3T^{2} \)
61 \( 1 - 82T + 3.72e3T^{2} \)
67 \( 1 + 24iT - 4.48e3T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 + 74iT - 5.32e3T^{2} \)
79 \( 1 + 138T + 6.24e3T^{2} \)
83 \( 1 + 93.9T + 6.88e3T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 - 166iT - 9.40e3T^{2} \)
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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

−14.06902651332128290111432432242, −13.32414707109295385458927180571, −12.31657548681470100757168704680, −11.53350113638795515282405894301, −9.347234988067112550107610244410, −8.594793722974099512644820849458, −6.88725113423520709060004447181, −5.69209964193781473695484719826, −4.00876422713434947061335680192, −2.48672149529654840341633430622, 3.07297585787477368925610568184, 4.24628357171964507243442624048, 5.39194695345011453929921597559, 7.27020219943647043171970581362, 8.695916637809971321562132067150, 9.957489967599238255946509188327, 11.02757305754365312200304710091, 12.68943509124999069698359647709, 13.37829718466998463341785714974, 14.37068391742290327613419582952

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