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Properties

Label 2-75-5.4-c17-0-21
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 105. i·2-s + 6.56e3i·3-s + 1.19e5·4-s − 6.92e5·6-s + 2.39e7i·7-s + 2.64e7i·8-s − 4.30e7·9-s − 5.09e8·11-s + 7.86e8i·12-s + 4.71e9i·13-s − 2.52e9·14-s + 1.29e10·16-s + 4.44e10i·17-s − 4.54e9i·18-s + 4.25e10·19-s + ⋯
L(s)  = 1  + 0.291i·2-s + 0.577i·3-s + 0.915·4-s − 0.168·6-s + 1.56i·7-s + 0.558i·8-s − 0.333·9-s − 0.716·11-s + 0.528i·12-s + 1.60i·13-s − 0.457·14-s + 0.752·16-s + 1.54i·17-s − 0.0971i·18-s + 0.574·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(2.508677973\)
\(L(\frac12)\) \(\approx\) \(2.508677973\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 6.56e3iT \)
5 \( 1 \)
good2 \( 1 - 105. iT - 1.31e5T^{2} \)
7 \( 1 - 2.39e7iT - 2.32e14T^{2} \)
11 \( 1 + 5.09e8T + 5.05e17T^{2} \)
13 \( 1 - 4.71e9iT - 8.65e18T^{2} \)
17 \( 1 - 4.44e10iT - 8.27e20T^{2} \)
19 \( 1 - 4.25e10T + 5.48e21T^{2} \)
23 \( 1 - 4.70e11iT - 1.41e23T^{2} \)
29 \( 1 + 3.15e12T + 7.25e24T^{2} \)
31 \( 1 - 3.46e12T + 2.25e25T^{2} \)
37 \( 1 + 3.18e13iT - 4.56e26T^{2} \)
41 \( 1 - 8.13e13T + 2.61e27T^{2} \)
43 \( 1 + 1.24e14iT - 5.87e27T^{2} \)
47 \( 1 - 7.05e13iT - 2.66e28T^{2} \)
53 \( 1 - 2.51e14iT - 2.05e29T^{2} \)
59 \( 1 - 6.39e14T + 1.27e30T^{2} \)
61 \( 1 + 4.96e13T + 2.24e30T^{2} \)
67 \( 1 - 2.75e15iT - 1.10e31T^{2} \)
71 \( 1 - 2.50e15T + 2.96e31T^{2} \)
73 \( 1 - 8.85e14iT - 4.74e31T^{2} \)
79 \( 1 - 6.85e13T + 1.81e32T^{2} \)
83 \( 1 - 3.38e16iT - 4.21e32T^{2} \)
89 \( 1 - 3.37e16T + 1.37e33T^{2} \)
97 \( 1 + 6.48e16iT - 5.95e33T^{2} \)
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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

−11.68570204858601322392259095683, −10.91619065849252336536892370398, −9.538980795780019933539854939508, −8.619460121600885484173959774027, −7.40932931850788025440962477724, −6.01009034871899616961011131889, −5.44911598262277834508412415450, −3.83702804165021191562603866511, −2.47930619717334067538254912113, −1.76601857712670105248704617387, 0.52152055728567989124971167506, 0.984488203538739914833421419143, 2.53803765802789144191743955701, 3.30767126327739125256275581772, 4.93359487275777056436062158175, 6.33004385670444114370674033182, 7.42812374339715194708005283989, 7.86519202130812514274940919207, 9.903537870620825947160989105141, 10.66192631682548445759312732996

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