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Properties

Label 2-75-5.4-c17-0-10
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  + 659. i·2-s − 6.56e3i·3-s − 3.03e5·4-s + 4.32e6·6-s + 1.59e6i·7-s − 1.13e8i·8-s − 4.30e7·9-s − 4.47e8·11-s + 1.99e9i·12-s + 2.48e9i·13-s − 1.05e9·14-s + 3.50e10·16-s − 2.48e10i·17-s − 2.83e10i·18-s − 8.23e10·19-s + ⋯
L(s)  = 1  + 1.82i·2-s − 0.577i·3-s − 2.31·4-s + 1.05·6-s + 0.104i·7-s − 2.39i·8-s − 0.333·9-s − 0.629·11-s + 1.33i·12-s + 0.844i·13-s − 0.190·14-s + 2.04·16-s − 0.864i·17-s − 0.606i·18-s − 1.11·19-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

\(L(9)\) \(\approx\) \(1.006082740\)
\(L(\frac12)\) \(\approx\) \(1.006082740\)
\(L(\frac{19}{2})\) 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 \( 1 - 659. iT - 1.31e5T^{2} \)
7 \( 1 - 1.59e6iT - 2.32e14T^{2} \)
11 \( 1 + 4.47e8T + 5.05e17T^{2} \)
13 \( 1 - 2.48e9iT - 8.65e18T^{2} \)
17 \( 1 + 2.48e10iT - 8.27e20T^{2} \)
19 \( 1 + 8.23e10T + 5.48e21T^{2} \)
23 \( 1 + 6.43e11iT - 1.41e23T^{2} \)
29 \( 1 - 9.82e11T + 7.25e24T^{2} \)
31 \( 1 - 3.28e12T + 2.25e25T^{2} \)
37 \( 1 - 2.63e13iT - 4.56e26T^{2} \)
41 \( 1 + 3.33e13T + 2.61e27T^{2} \)
43 \( 1 + 9.83e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.62e14iT - 2.66e28T^{2} \)
53 \( 1 - 1.40e14iT - 2.05e29T^{2} \)
59 \( 1 - 9.80e13T + 1.27e30T^{2} \)
61 \( 1 - 1.37e15T + 2.24e30T^{2} \)
67 \( 1 + 1.85e15iT - 1.10e31T^{2} \)
71 \( 1 + 6.17e15T + 2.96e31T^{2} \)
73 \( 1 - 1.30e16iT - 4.74e31T^{2} \)
79 \( 1 + 1.27e16T + 1.81e32T^{2} \)
83 \( 1 - 1.42e16iT - 4.21e32T^{2} \)
89 \( 1 - 3.77e16T + 1.37e33T^{2} \)
97 \( 1 - 1.09e17iT - 5.95e33T^{2} \)
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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

−11.98500051426213947680452457315, −10.26311376582517362225135979018, −8.838584302886072510845797787163, −8.232671686702587797302318607953, −6.99859491345439352981814204849, −6.45912564168433579462045611864, −5.23323194240841289462164232107, −4.28422158606344547017192508413, −2.42467515805633998231852334885, −0.62900685106483919724106571850, 0.32882345626207830342872968180, 1.56040752652591603997548901871, 2.69609969212987938840225589975, 3.61433023402453684535402310627, 4.58157400579155726835990359301, 5.73757433226548859528569932830, 7.958358283855423296057460732817, 9.011773010729746636429716360863, 10.12456718705718360758117790856, 10.64756458871212103411936605807

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