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Properties

Label 2-75-5.4-c17-0-43
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  − 352. i·2-s + 6.56e3i·3-s + 6.49e3·4-s + 2.31e6·6-s − 1.07e7i·7-s − 4.85e7i·8-s − 4.30e7·9-s + 1.00e9·11-s + 4.25e7i·12-s + 3.43e9i·13-s − 3.77e9·14-s − 1.62e10·16-s − 4.81e10i·17-s + 1.51e10i·18-s − 8.08e9·19-s + ⋯
L(s)  = 1  − 0.974i·2-s + 0.577i·3-s + 0.0495·4-s + 0.562·6-s − 0.701i·7-s − 1.02i·8-s − 0.333·9-s + 1.41·11-s + 0.0285i·12-s + 1.16i·13-s − 0.683·14-s − 0.948·16-s − 1.67i·17-s + 0.324i·18-s − 0.109·19-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

\(L(9)\) \(\approx\) \(2.235425346\)
\(L(\frac12)\) \(\approx\) \(2.235425346\)
\(L(\frac{19}{2})\) 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 \( 1 + 352. iT - 1.31e5T^{2} \)
7 \( 1 + 1.07e7iT - 2.32e14T^{2} \)
11 \( 1 - 1.00e9T + 5.05e17T^{2} \)
13 \( 1 - 3.43e9iT - 8.65e18T^{2} \)
17 \( 1 + 4.81e10iT - 8.27e20T^{2} \)
19 \( 1 + 8.08e9T + 5.48e21T^{2} \)
23 \( 1 + 3.08e11iT - 1.41e23T^{2} \)
29 \( 1 + 7.97e11T + 7.25e24T^{2} \)
31 \( 1 + 3.17e12T + 2.25e25T^{2} \)
37 \( 1 + 1.61e13iT - 4.56e26T^{2} \)
41 \( 1 - 7.63e13T + 2.61e27T^{2} \)
43 \( 1 - 1.29e14iT - 5.87e27T^{2} \)
47 \( 1 - 2.44e14iT - 2.66e28T^{2} \)
53 \( 1 + 8.60e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.21e14T + 1.27e30T^{2} \)
61 \( 1 + 4.27e14T + 2.24e30T^{2} \)
67 \( 1 - 1.16e15iT - 1.10e31T^{2} \)
71 \( 1 - 3.01e15T + 2.96e31T^{2} \)
73 \( 1 + 7.32e15iT - 4.74e31T^{2} \)
79 \( 1 - 7.07e15T + 1.81e32T^{2} \)
83 \( 1 + 1.18e16iT - 4.21e32T^{2} \)
89 \( 1 + 5.69e16T + 1.37e33T^{2} \)
97 \( 1 - 8.11e16iT - 5.95e33T^{2} \)
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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

−11.02476642187179165952153209493, −9.657449745001240350767963051749, −9.187687621063132968884534312728, −7.27560662094723130768558990119, −6.38717901497315335080585632159, −4.52010662412983167637659687028, −3.84604204301378339362943209623, −2.65583871197156876998159255470, −1.44620741513241312774020215961, −0.43724592555007185778693894566, 1.24650497383917139468158563852, 2.28169363605382948290548847971, 3.73279905764233663821752044190, 5.53425468375362852712937386025, 6.10551342310537691503291254830, 7.17268338920439806687947856012, 8.193032285474310601350376620810, 9.032062360642983905134843079456, 10.69436717051317056920294460668, 11.82493946015150196025547986573

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