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Properties

Label 2-75-5.4-c21-0-56
Degree $2$
Conductor $75$
Sign $-0.894 - 0.447i$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27e3i·2-s − 5.90e4i·3-s + 4.79e5·4-s − 7.51e7·6-s + 6.32e8i·7-s − 3.27e9i·8-s − 3.48e9·9-s + 5.97e10·11-s − 2.83e10i·12-s − 7.38e11i·13-s + 8.03e11·14-s − 3.16e12·16-s − 8.35e12i·17-s + 4.43e12i·18-s − 4.19e13·19-s + ⋯
L(s)  = 1  − 0.878i·2-s − 0.577i·3-s + 0.228·4-s − 0.507·6-s + 0.845i·7-s − 1.07i·8-s − 0.333·9-s + 0.694·11-s − 0.131i·12-s − 1.48i·13-s + 0.742·14-s − 0.719·16-s − 1.00i·17-s + 0.292i·18-s − 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.242449647\)
\(L(\frac12)\) \(\approx\) \(2.242449647\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.90e4iT \)
5 \( 1 \)
good2 \( 1 + 1.27e3iT - 2.09e6T^{2} \)
7 \( 1 - 6.32e8iT - 5.58e17T^{2} \)
11 \( 1 - 5.97e10T + 7.40e21T^{2} \)
13 \( 1 + 7.38e11iT - 2.47e23T^{2} \)
17 \( 1 + 8.35e12iT - 6.90e25T^{2} \)
19 \( 1 + 4.19e13T + 7.14e26T^{2} \)
23 \( 1 + 4.48e13iT - 3.94e28T^{2} \)
29 \( 1 - 2.76e15T + 5.13e30T^{2} \)
31 \( 1 - 8.36e15T + 2.08e31T^{2} \)
37 \( 1 + 1.77e16iT - 8.55e32T^{2} \)
41 \( 1 - 1.45e17T + 7.38e33T^{2} \)
43 \( 1 + 1.24e17iT - 2.00e34T^{2} \)
47 \( 1 - 4.28e17iT - 1.30e35T^{2} \)
53 \( 1 - 4.77e17iT - 1.62e36T^{2} \)
59 \( 1 + 1.61e18T + 1.54e37T^{2} \)
61 \( 1 + 3.76e18T + 3.10e37T^{2} \)
67 \( 1 + 2.81e18iT - 2.22e38T^{2} \)
71 \( 1 - 1.00e19T + 7.52e38T^{2} \)
73 \( 1 - 1.72e19iT - 1.34e39T^{2} \)
79 \( 1 - 3.28e19T + 7.08e39T^{2} \)
83 \( 1 + 3.05e17iT - 1.99e40T^{2} \)
89 \( 1 + 2.34e20T + 8.65e40T^{2} \)
97 \( 1 + 5.92e20iT - 5.27e41T^{2} \)
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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

−10.26949938714632504980686518589, −9.065795659541318787051725766843, −7.972715030782159659530768488031, −6.69224736344413004036323096692, −5.86240543607005187397095327084, −4.35021964433810738816001787369, −2.84952277548593587170661003060, −2.47537815326291899209478447555, −1.17307038601411434892283850991, −0.40058697550043657138456475694, 1.25251227757390832482223488218, 2.42348668898736925595271242704, 4.00481222297773563482225710949, 4.62767890489364007231218704559, 6.29640882822115764512252909217, 6.63739855544251972700228819658, 8.015269581128377723741692581802, 8.901128781049873104104098314877, 10.21801295484969687473766273490, 11.11290954418206893715961523766

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