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Properties

Label 2-5-5.3-c16-0-1
Degree $2$
Conductor $5$
Sign $0.376 - 0.926i$
Analytic cond. $8.11622$
Root an. cond. $2.84889$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−205. + 205. i)2-s + (−8.38e3 − 8.38e3i)3-s − 1.85e4i·4-s + (−6.50e4 − 3.85e5i)5-s + 3.44e6·6-s + (−4.20e6 + 4.20e6i)7-s + (−9.62e6 − 9.62e6i)8-s + 9.77e7i·9-s + (9.23e7 + 6.56e7i)10-s + 1.64e8·11-s + (−1.55e8 + 1.55e8i)12-s + (2.05e8 + 2.05e8i)13-s − 1.72e9i·14-s + (−2.68e9 + 3.77e9i)15-s + 5.16e9·16-s + (2.00e9 − 2.00e9i)17-s + ⋯
L(s)  = 1  + (−0.801 + 0.801i)2-s + (−1.27 − 1.27i)3-s − 0.283i·4-s + (−0.166 − 0.986i)5-s + 2.04·6-s + (−0.728 + 0.728i)7-s + (−0.573 − 0.573i)8-s + 2.27i·9-s + (0.923 + 0.656i)10-s + 0.765·11-s + (−0.362 + 0.362i)12-s + (0.251 + 0.251i)13-s − 1.16i·14-s + (−1.04 + 1.47i)15-s + 1.20·16-s + (0.287 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(8.11622\)
Root analytic conductor: \(2.84889\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :8),\ 0.376 - 0.926i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.332202 + 0.223518i\)
\(L(\frac12)\) \(\approx\) \(0.332202 + 0.223518i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.50e4 + 3.85e5i)T \)
good2 \( 1 + (205. - 205. i)T - 6.55e4iT^{2} \)
3 \( 1 + (8.38e3 + 8.38e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (4.20e6 - 4.20e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 1.64e8T + 4.59e16T^{2} \)
13 \( 1 + (-2.05e8 - 2.05e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-2.00e9 + 2.00e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 6.44e9iT - 2.88e20T^{2} \)
23 \( 1 + (-2.07e10 - 2.07e10i)T + 6.13e21iT^{2} \)
29 \( 1 - 7.06e11iT - 2.50e23T^{2} \)
31 \( 1 + 3.56e11T + 7.27e23T^{2} \)
37 \( 1 + (-2.11e11 + 2.11e11i)T - 1.23e25iT^{2} \)
41 \( 1 - 8.10e12T + 6.37e25T^{2} \)
43 \( 1 + (6.40e12 + 6.40e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (2.97e12 - 2.97e12i)T - 5.66e26iT^{2} \)
53 \( 1 + (-8.09e13 - 8.09e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 1.64e13iT - 2.15e28T^{2} \)
61 \( 1 - 1.82e14T + 3.67e28T^{2} \)
67 \( 1 + (4.64e14 - 4.64e14i)T - 1.64e29iT^{2} \)
71 \( 1 + 9.33e14T + 4.16e29T^{2} \)
73 \( 1 + (-8.72e14 - 8.72e14i)T + 6.50e29iT^{2} \)
79 \( 1 + 5.24e14iT - 2.30e30T^{2} \)
83 \( 1 + (-1.59e15 - 1.59e15i)T + 5.07e30iT^{2} \)
89 \( 1 + 5.35e15iT - 1.54e31T^{2} \)
97 \( 1 + (6.01e15 - 6.01e15i)T - 6.14e31iT^{2} \)
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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

−19.32166379312465613981493318225, −18.14621149810066406636369131253, −16.95353989655844778142450532563, −16.10459335528811840731195319439, −12.87811192065241421099429062486, −11.91522046664174144927890698724, −8.937609186991686365952893728785, −7.13528559000091076648955769744, −5.78417264509509327307395924178, −0.996232177040116314764800138517, 0.41931919893275409725806967857, 3.66519258132799312901559417841, 6.18369301974784976561432232958, 9.737361915736190198564016379712, 10.57799134037337712394631246827, 11.66879319787202060165138365464, 14.93676856051539899374808162225, 16.61281428389725496005597227231, 17.86098035679897946402536602099, 19.46011054161238149142265189172

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