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Properties

Label 2-75-5.4-c3-0-0
Degree $2$
Conductor $75$
Sign $-0.447 + 0.894i$
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  + 5.35i·2-s + 3i·3-s − 20.7·4-s − 16.0·6-s + 4.43i·7-s − 68.1i·8-s − 9·9-s − 3.43·11-s − 62.1i·12-s + 78.7i·13-s − 23.7·14-s + 199.·16-s − 53.1i·17-s − 48.2i·18-s − 20.4·19-s + ⋯
L(s)  = 1  + 1.89i·2-s + 0.577i·3-s − 2.58·4-s − 1.09·6-s + 0.239i·7-s − 3.01i·8-s − 0.333·9-s − 0.0941·11-s − 1.49i·12-s + 1.67i·13-s − 0.453·14-s + 3.11·16-s − 0.758i·17-s − 0.631i·18-s − 0.246·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.488234 - 0.789980i\)
\(L(\frac12)\) \(\approx\) \(0.488234 - 0.789980i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
good2 \( 1 - 5.35iT - 8T^{2} \)
7 \( 1 - 4.43iT - 343T^{2} \)
11 \( 1 + 3.43T + 1.33e3T^{2} \)
13 \( 1 - 78.7iT - 2.19e3T^{2} \)
17 \( 1 + 53.1iT - 4.91e3T^{2} \)
19 \( 1 + 20.4T + 6.85e3T^{2} \)
23 \( 1 - 118. iT - 1.21e4T^{2} \)
29 \( 1 + 168.T + 2.43e4T^{2} \)
31 \( 1 + 61.0T + 2.97e4T^{2} \)
37 \( 1 - 246. iT - 5.06e4T^{2} \)
41 \( 1 - 422.T + 6.89e4T^{2} \)
43 \( 1 - 362. iT - 7.95e4T^{2} \)
47 \( 1 - 170. iT - 1.03e5T^{2} \)
53 \( 1 + 546. iT - 1.48e5T^{2} \)
59 \( 1 - 216.T + 2.05e5T^{2} \)
61 \( 1 - 130.T + 2.26e5T^{2} \)
67 \( 1 - 614. iT - 3.00e5T^{2} \)
71 \( 1 - 324.T + 3.57e5T^{2} \)
73 \( 1 + 88.8iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 758. iT - 5.71e5T^{2} \)
89 \( 1 + 195.T + 7.04e5T^{2} \)
97 \( 1 + 521iT - 9.12e5T^{2} \)
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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

−14.88664515975937012920470267195, −14.16694804937160182653144285002, −13.17779420937502335592996015560, −11.50763584985851991600889436103, −9.605634191848528638240664511418, −9.016409255420323535786135507499, −7.66458197256976229522059133330, −6.52146348711616726289397076246, −5.28289586804641091121899599646, −4.11145855561427279280682371903, 0.61540808657969934405038730369, 2.36588921557641618776462538371, 3.82677162278797454927420308674, 5.54960574764708893946258587933, 7.85261574968612288806807483214, 9.004487738229617867230255078958, 10.41762158446238611407580305180, 10.95399872720864694846374271157, 12.44186265810578522469350543097, 12.78760111934190969300523945252

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