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Properties

Label 2-75-15.14-c2-0-8
Degree $2$
Conductor $75$
Sign $-0.447 + 0.894i$
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  − 3i·3-s − 4·4-s − 11i·7-s − 9·9-s + 12i·12-s − i·13-s + 16·16-s + 37·19-s − 33·21-s + 27i·27-s + 44i·28-s − 13·31-s + 36·36-s − 26i·37-s − 3·39-s + ⋯
L(s)  = 1  − i·3-s − 4-s − 1.57i·7-s − 9-s + i·12-s − 0.0769i·13-s + 16-s + 1.94·19-s − 1.57·21-s + i·27-s + 1.57i·28-s − 0.419·31-s + 36-s − 0.702i·37-s − 0.0769·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 75,\ (\ :1),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.474625 - 0.767959i\)
\(L(\frac12)\) \(\approx\) \(0.474625 - 0.767959i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
good2 \( 1 + 4T^{2} \)
7 \( 1 + 11iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 - 37T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 13T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 61iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 47T + 3.72e3T^{2} \)
67 \( 1 - 109iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46iT - 5.32e3T^{2} \)
79 \( 1 - 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 169iT - 9.40e3T^{2} \)
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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

−13.76165706987377172305799302461, −13.17807981962901799163044293868, −11.95124334429892887341315238770, −10.58389939498170442206218058872, −9.341114723153192091921942940831, −7.929308817635791345093554598675, −7.09129825850865438987549049126, −5.35047507057977091293151893717, −3.66397516908791709869342815243, −0.859229389880520879695665431533, 3.20203463841436982829307878115, 4.89295320974082660446728005919, 5.75462090382639367735554950455, 8.168283394102852171471206320964, 9.203868736757563951448684387432, 9.788244000507150098621056394312, 11.39499158191839761953432580722, 12.37421305010241471478071671033, 13.76713982750329349389723601579, 14.72730958661177218596959692707

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