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Properties

Label 2-5-5.3-c4-0-0
Degree $2$
Conductor $5$
Sign $0.995 - 0.0898i$
Analytic cond. $0.516849$
Root an. cond. $0.718922$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−6 − 6i)3-s + 14i·4-s + (20 − 15i)5-s + 12·6-s + (−26 + 26i)7-s + (−30 − 30i)8-s − 9i·9-s + (−5 + 35i)10-s − 8·11-s + (84 − 84i)12-s + (139 + 139i)13-s − 52i·14-s + (−210 − 30i)15-s − 164·16-s + (−1 + i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.250i)2-s + (−0.666 − 0.666i)3-s + 0.875i·4-s + (0.800 − 0.599i)5-s + 0.333·6-s + (−0.530 + 0.530i)7-s + (−0.468 − 0.468i)8-s − 0.111i·9-s + (−0.0500 + 0.350i)10-s − 0.0661·11-s + (0.583 − 0.583i)12-s + (0.822 + 0.822i)13-s − 0.265i·14-s + (−0.933 − 0.133i)15-s − 0.640·16-s + (−0.00346 + 0.00346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.995 - 0.0898i$
Analytic conductor: \(0.516849\)
Root analytic conductor: \(0.718922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :2),\ 0.995 - 0.0898i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.669000 + 0.0301007i\)
\(L(\frac12)\) \(\approx\) \(0.669000 + 0.0301007i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-20 + 15i)T \)
good2 \( 1 + (1 - i)T - 16iT^{2} \)
3 \( 1 + (6 + 6i)T + 81iT^{2} \)
7 \( 1 + (26 - 26i)T - 2.40e3iT^{2} \)
11 \( 1 + 8T + 1.46e4T^{2} \)
13 \( 1 + (-139 - 139i)T + 2.85e4iT^{2} \)
17 \( 1 + (1 - i)T - 8.35e4iT^{2} \)
19 \( 1 + 180iT - 1.30e5T^{2} \)
23 \( 1 + (166 + 166i)T + 2.79e5iT^{2} \)
29 \( 1 - 480iT - 7.07e5T^{2} \)
31 \( 1 - 572T + 9.23e5T^{2} \)
37 \( 1 + (251 - 251i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.47e3 - 1.47e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-2.47e3 + 2.47e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (3.33e3 + 3.33e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.66e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.59e3T + 1.38e7T^{2} \)
67 \( 1 + (-874 + 874i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.06e3T + 2.54e7T^{2} \)
73 \( 1 + (791 + 791i)T + 2.83e7iT^{2} \)
79 \( 1 + 9.12e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.65e3 - 5.65e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 2.16e3iT - 6.27e7T^{2} \)
97 \( 1 + (6.55e3 - 6.55e3i)T - 8.85e7iT^{2} \)
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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

−23.89913624452159933516741890958, −22.15309634886302607912586360934, −20.88647769646167381974628978295, −18.50275335165158129876739462903, −17.40897162649728375522062915889, −16.16030001239991370174501813980, −13.23394830568435968840208799345, −12.03131972356520577702595263087, −8.992004666314940419028087030184, −6.43446901279239483854321391270, 5.87345398363083830302364819598, 9.959603234283732465813800828395, 10.84818845483526284000633729910, 13.78514932864529845373034703274, 15.64547499604769225484994205118, 17.41568124951225360560227193675, 18.91495198703454935137597208810, 20.63801047044014262264701318701, 22.28910821114774274241655878508, 23.26434389688325556266865988942

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