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Properties

Label 2-75-5.4-c17-0-29
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  + 213. i·2-s − 6.56e3i·3-s + 8.52e4·4-s + 1.40e6·6-s − 7.24e6i·7-s + 4.62e7i·8-s − 4.30e7·9-s + 4.10e7·11-s − 5.59e8i·12-s + 1.35e9i·13-s + 1.54e9·14-s + 1.27e9·16-s − 1.84e10i·17-s − 9.21e9i·18-s − 1.42e9·19-s + ⋯
L(s)  = 1  + 0.590i·2-s − 0.577i·3-s + 0.650·4-s + 0.341·6-s − 0.474i·7-s + 0.975i·8-s − 0.333·9-s + 0.0577·11-s − 0.375i·12-s + 0.461i·13-s + 0.280·14-s + 0.0742·16-s − 0.639i·17-s − 0.196i·18-s − 0.0192·19-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

\(L(9)\) \(\approx\) \(2.827108225\)
\(L(\frac12)\) \(\approx\) \(2.827108225\)
\(L(\frac{19}{2})\) 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 \( 1 - 213. iT - 1.31e5T^{2} \)
7 \( 1 + 7.24e6iT - 2.32e14T^{2} \)
11 \( 1 - 4.10e7T + 5.05e17T^{2} \)
13 \( 1 - 1.35e9iT - 8.65e18T^{2} \)
17 \( 1 + 1.84e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.42e9T + 5.48e21T^{2} \)
23 \( 1 + 1.96e10iT - 1.41e23T^{2} \)
29 \( 1 - 3.22e12T + 7.25e24T^{2} \)
31 \( 1 + 1.93e12T + 2.25e25T^{2} \)
37 \( 1 - 2.47e13iT - 4.56e26T^{2} \)
41 \( 1 - 1.91e13T + 2.61e27T^{2} \)
43 \( 1 + 9.80e12iT - 5.87e27T^{2} \)
47 \( 1 + 6.08e13iT - 2.66e28T^{2} \)
53 \( 1 - 3.72e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.78e15T + 1.27e30T^{2} \)
61 \( 1 - 1.98e15T + 2.24e30T^{2} \)
67 \( 1 - 4.22e15iT - 1.10e31T^{2} \)
71 \( 1 + 1.65e15T + 2.96e31T^{2} \)
73 \( 1 + 1.26e16iT - 4.74e31T^{2} \)
79 \( 1 - 1.63e16T + 1.81e32T^{2} \)
83 \( 1 + 2.45e16iT - 4.21e32T^{2} \)
89 \( 1 - 4.52e15T + 1.37e33T^{2} \)
97 \( 1 + 3.59e16iT - 5.95e33T^{2} \)
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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

−11.41687189171336426685603711023, −10.31147853938572630335466465256, −8.808598079371899280359518135054, −7.67690977843958919161587694342, −6.90935303484072842665587126121, −6.02397288267478346852173467834, −4.71299239705094662650102976557, −3.07702500637682254146496997752, −1.95911979106194707974779140393, −0.810780481095365103245751076298, 0.70753587382540807035841414317, 2.01179183840043152774476804340, 2.99540273053174287631368803312, 4.03688596487773881876810105182, 5.48217850871892782537670337244, 6.55378784730633953089350072979, 7.935120922271313936965924301521, 9.197180603957869188795193983018, 10.27694857029590651372201762597, 11.01868563401728704347060230159

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