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Properties

Label 2-5-5.3-c10-0-3
Degree $2$
Conductor $5$
Sign $-0.544 + 0.838i$
Analytic cond. $3.17678$
Root an. cond. $1.78235$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.8 − 18.8i)2-s + (−273. − 273. i)3-s + 314. i·4-s + (71.2 − 3.12e3i)5-s − 1.02e4·6-s + (1.26e4 − 1.26e4i)7-s + (2.52e4 + 2.52e4i)8-s + 9.00e4i·9-s + (−5.75e4 − 6.02e4i)10-s + 5.73e4·11-s + (8.57e4 − 8.57e4i)12-s + (−1.92e5 − 1.92e5i)13-s − 4.76e5i·14-s + (−8.72e5 + 8.33e5i)15-s + 6.28e5·16-s + (4.20e5 − 4.20e5i)17-s + ⋯
L(s)  = 1  + (0.588 − 0.588i)2-s + (−1.12 − 1.12i)3-s + 0.306i·4-s + (0.0228 − 0.999i)5-s − 1.32·6-s + (0.752 − 0.752i)7-s + (0.769 + 0.769i)8-s + 1.52i·9-s + (−0.575 − 0.602i)10-s + 0.356·11-s + (0.344 − 0.344i)12-s + (−0.519 − 0.519i)13-s − 0.885i·14-s + (−1.14 + 1.09i)15-s + 0.599·16-s + (0.295 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(3.17678\)
Root analytic conductor: \(1.78235\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :5),\ -0.544 + 0.838i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.662651 - 1.22107i\)
\(L(\frac12)\) \(\approx\) \(0.662651 - 1.22107i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-71.2 + 3.12e3i)T \)
good2 \( 1 + (-18.8 + 18.8i)T - 1.02e3iT^{2} \)
3 \( 1 + (273. + 273. i)T + 5.90e4iT^{2} \)
7 \( 1 + (-1.26e4 + 1.26e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 5.73e4T + 2.59e10T^{2} \)
13 \( 1 + (1.92e5 + 1.92e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-4.20e5 + 4.20e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 4.70e5iT - 6.13e12T^{2} \)
23 \( 1 + (-7.12e6 - 7.12e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 9.85e6iT - 4.20e14T^{2} \)
31 \( 1 - 4.06e7T + 8.19e14T^{2} \)
37 \( 1 + (-5.42e7 + 5.42e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 7.34e7T + 1.34e16T^{2} \)
43 \( 1 + (1.30e7 + 1.30e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (9.53e7 - 9.53e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (1.58e8 + 1.58e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 6.48e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.02e9T + 7.13e17T^{2} \)
67 \( 1 + (-1.15e9 + 1.15e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 5.90e8T + 3.25e18T^{2} \)
73 \( 1 + (1.16e9 + 1.16e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 4.58e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.35e9 + 1.35e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 6.88e9iT - 3.11e19T^{2} \)
97 \( 1 + (3.03e8 - 3.03e8i)T - 7.37e19iT^{2} \)
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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

−21.19867290474215669621386583393, −19.79049218198133604824368586209, −17.56526569485216915365796940405, −16.89741329802620254809849334985, −13.56385937903915327138759703659, −12.42489116726887892306192214897, −11.31515975096187011991413676658, −7.64178017981663661456941051604, −4.97480453955042097089288326518, −1.18256320130485293640642700135, 4.73960818724259101521030999970, 6.30519265356449029829645239832, 10.12578871449839639334283628742, 11.53357107600279758154479802758, 14.54990340651987512823346241394, 15.40735219505773327795681683668, 17.00641944497067908090061578233, 18.77142320102722425733206346519, 21.41922388711181318887147986969, 22.33046288344419681360101802983

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