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Properties

Label 2-5-5.3-c26-0-10
Degree $2$
Conductor $5$
Sign $-0.437 + 0.899i$
Analytic cond. $21.4146$
Root an. cond. $4.62759$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11e4 − 1.11e4i)2-s + (1.76e6 + 1.76e6i)3-s − 1.80e8i·4-s + (−1.23e8 − 1.21e9i)5-s + 3.92e10·6-s + (1.33e10 − 1.33e10i)7-s + (−1.25e12 − 1.25e12i)8-s + 3.67e12i·9-s + (−1.48e13 − 1.21e13i)10-s + 9.56e12·11-s + (3.17e14 − 3.17e14i)12-s + (4.32e13 + 4.32e13i)13-s − 2.97e14i·14-s + (1.92e15 − 2.35e15i)15-s − 1.58e16·16-s + (7.63e15 − 7.63e15i)17-s + ⋯
L(s)  = 1  + (1.35 − 1.35i)2-s + (1.10 + 1.10i)3-s − 2.68i·4-s + (−0.100 − 0.994i)5-s + 3.00·6-s + (0.138 − 0.138i)7-s + (−2.28 − 2.28i)8-s + 1.44i·9-s + (−1.48 − 1.21i)10-s + 0.276·11-s + (2.96 − 2.96i)12-s + (0.142 + 0.142i)13-s − 0.374i·14-s + (0.988 − 1.21i)15-s − 3.52·16-s + (0.770 − 0.770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(2.90044 - 4.63562i\)
\(L(\frac12)\) \(\approx\) \(2.90044 - 4.63562i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.23e8 + 1.21e9i)T \)
good2 \( 1 + (-1.11e4 + 1.11e4i)T - 6.71e7iT^{2} \)
3 \( 1 + (-1.76e6 - 1.76e6i)T + 2.54e12iT^{2} \)
7 \( 1 + (-1.33e10 + 1.33e10i)T - 9.38e21iT^{2} \)
11 \( 1 - 9.56e12T + 1.19e27T^{2} \)
13 \( 1 + (-4.32e13 - 4.32e13i)T + 9.17e28iT^{2} \)
17 \( 1 + (-7.63e15 + 7.63e15i)T - 9.81e31iT^{2} \)
19 \( 1 - 3.69e15iT - 1.76e33T^{2} \)
23 \( 1 + (-6.37e16 - 6.37e16i)T + 2.54e35iT^{2} \)
29 \( 1 - 1.61e19iT - 1.05e38T^{2} \)
31 \( 1 - 2.82e19T + 5.96e38T^{2} \)
37 \( 1 + (1.75e20 - 1.75e20i)T - 5.93e40iT^{2} \)
41 \( 1 + 5.21e20T + 8.55e41T^{2} \)
43 \( 1 + (-1.20e21 - 1.20e21i)T + 2.95e42iT^{2} \)
47 \( 1 + (-5.45e21 + 5.45e21i)T - 2.98e43iT^{2} \)
53 \( 1 + (8.65e20 + 8.65e20i)T + 6.77e44iT^{2} \)
59 \( 1 - 2.83e22iT - 1.10e46T^{2} \)
61 \( 1 - 2.33e23T + 2.62e46T^{2} \)
67 \( 1 + (7.52e23 - 7.52e23i)T - 3.00e47iT^{2} \)
71 \( 1 + 7.35e23T + 1.35e48T^{2} \)
73 \( 1 + (7.67e23 + 7.67e23i)T + 2.79e48iT^{2} \)
79 \( 1 + 5.55e24iT - 2.17e49T^{2} \)
83 \( 1 + (-3.80e24 - 3.80e24i)T + 7.87e49iT^{2} \)
89 \( 1 + 5.72e24iT - 4.83e50T^{2} \)
97 \( 1 + (4.50e25 - 4.50e25i)T - 4.52e51iT^{2} \)
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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

−15.94409864470385458970195104468, −14.58353317532009389065586805139, −13.54287034164059881731010998939, −11.96867954763487714586138002417, −10.22702515889826719124609259489, −9.023374047447705867764113411889, −5.13914544886561342093665266136, −4.11362464869111441844919276000, −2.95925595035771079118298388244, −1.26389029503922792865635226052, 2.49792082951214846318648494422, 3.72191004958016663294634157281, 6.11577838833018512543503015093, 7.28132490109756215040783689791, 8.276217097464873011197608379052, 12.15085352773890772559399255867, 13.54127458911636465497284888950, 14.40237566711127522439484941793, 15.36739489161838156562940323158, 17.48872024088933363568776757717

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