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Properties

Label 2-75-5.4-c17-0-41
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  − 346. i·2-s + 6.56e3i·3-s + 1.10e4·4-s + 2.27e6·6-s + 1.80e7i·7-s − 4.92e7i·8-s − 4.30e7·9-s − 1.45e8·11-s + 7.23e7i·12-s − 3.26e9i·13-s + 6.26e9·14-s − 1.56e10·16-s − 7.81e9i·17-s + 1.49e10i·18-s + 7.18e10·19-s + ⋯
L(s)  = 1  − 0.957i·2-s + 0.577i·3-s + 0.0841·4-s + 0.552·6-s + 1.18i·7-s − 1.03i·8-s − 0.333·9-s − 0.204·11-s + 0.0485i·12-s − 1.10i·13-s + 1.13·14-s − 0.908·16-s − 0.271i·17-s + 0.319i·18-s + 0.970·19-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

\(L(9)\) \(\approx\) \(1.255770781\)
\(L(\frac12)\) \(\approx\) \(1.255770781\)
\(L(\frac{19}{2})\) 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 \( 1 + 346. iT - 1.31e5T^{2} \)
7 \( 1 - 1.80e7iT - 2.32e14T^{2} \)
11 \( 1 + 1.45e8T + 5.05e17T^{2} \)
13 \( 1 + 3.26e9iT - 8.65e18T^{2} \)
17 \( 1 + 7.81e9iT - 8.27e20T^{2} \)
19 \( 1 - 7.18e10T + 5.48e21T^{2} \)
23 \( 1 - 1.67e11iT - 1.41e23T^{2} \)
29 \( 1 + 1.63e11T + 7.25e24T^{2} \)
31 \( 1 + 5.38e12T + 2.25e25T^{2} \)
37 \( 1 - 3.68e13iT - 4.56e26T^{2} \)
41 \( 1 + 7.52e12T + 2.61e27T^{2} \)
43 \( 1 + 5.73e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.42e14iT - 2.66e28T^{2} \)
53 \( 1 - 8.01e14iT - 2.05e29T^{2} \)
59 \( 1 + 9.64e14T + 1.27e30T^{2} \)
61 \( 1 + 1.29e15T + 2.24e30T^{2} \)
67 \( 1 + 4.71e15iT - 1.10e31T^{2} \)
71 \( 1 - 9.29e15T + 2.96e31T^{2} \)
73 \( 1 + 6.86e15iT - 4.74e31T^{2} \)
79 \( 1 + 1.21e16T + 1.81e32T^{2} \)
83 \( 1 + 3.23e16iT - 4.21e32T^{2} \)
89 \( 1 - 4.91e16T + 1.37e33T^{2} \)
97 \( 1 + 1.26e17iT - 5.95e33T^{2} \)
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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

−10.78937404703342016559208459222, −9.855743625524320767747406020460, −8.945790326943720501677083241073, −7.56210041349466735311224029013, −5.98241752213198458527040316097, −5.01046039966343453728592714115, −3.40727672662514515557199272915, −2.78615926810744916287178118543, −1.60305692580446779807407068797, −0.24157148347029332841601968967, 1.14195786633012488828626678627, 2.28028544094859737286009888415, 3.85176058422371472561984552908, 5.21303942947783564523447738849, 6.43583289089818584552884870565, 7.20578275912888039736064264619, 7.904684428938921594189158805972, 9.233396996658402933372903988523, 10.73503029583737941417769061113, 11.58896247683841562357953525526

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