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Properties

Label 2-5-5.4-c25-0-7
Degree $2$
Conductor $5$
Sign $0.997 + 0.0661i$
Analytic cond. $19.7998$
Root an. cond. $4.44970$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.36e3i·2-s + 5.31e5i·3-s − 5.40e7·4-s + (−5.44e8 − 3.61e7i)5-s − 4.97e9·6-s − 1.37e10i·7-s − 1.92e11i·8-s + 5.64e11·9-s + (3.37e11 − 5.09e12i)10-s − 9.26e12·11-s − 2.87e13i·12-s − 1.02e14i·13-s + 1.28e14·14-s + (1.91e13 − 2.89e14i)15-s − 1.55e13·16-s + 3.06e15i·17-s + ⋯
L(s)  = 1  + 1.61i·2-s + 0.577i·3-s − 1.61·4-s + (−0.997 − 0.0661i)5-s − 0.933·6-s − 0.375i·7-s − 0.988i·8-s + 0.666·9-s + (0.106 − 1.61i)10-s − 0.890·11-s − 0.930i·12-s − 1.21i·13-s + 0.607·14-s + (0.0381 − 0.576i)15-s − 0.0138·16-s + 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0661i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.997 + 0.0661i$
Analytic conductor: \(19.7998\)
Root analytic conductor: \(4.44970\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :25/2),\ 0.997 + 0.0661i)\)

Particular Values

\(L(13)\) \(\approx\) \(0.415399 - 0.0137501i\)
\(L(\frac12)\) \(\approx\) \(0.415399 - 0.0137501i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (5.44e8 + 3.61e7i)T \)
good2 \( 1 - 9.36e3iT - 3.35e7T^{2} \)
3 \( 1 - 5.31e5iT - 8.47e11T^{2} \)
7 \( 1 + 1.37e10iT - 1.34e21T^{2} \)
11 \( 1 + 9.26e12T + 1.08e26T^{2} \)
13 \( 1 + 1.02e14iT - 7.05e27T^{2} \)
17 \( 1 - 3.06e15iT - 5.77e30T^{2} \)
19 \( 1 - 5.00e15T + 9.30e31T^{2} \)
23 \( 1 + 9.11e16iT - 1.10e34T^{2} \)
29 \( 1 + 3.16e18T + 3.63e36T^{2} \)
31 \( 1 - 3.22e18T + 1.92e37T^{2} \)
37 \( 1 + 1.59e19iT - 1.60e39T^{2} \)
41 \( 1 + 8.10e19T + 2.08e40T^{2} \)
43 \( 1 + 3.97e20iT - 6.86e40T^{2} \)
47 \( 1 + 9.69e20iT - 6.34e41T^{2} \)
53 \( 1 + 5.88e21iT - 1.27e43T^{2} \)
59 \( 1 + 2.11e22T + 1.86e44T^{2} \)
61 \( 1 - 7.65e21T + 4.29e44T^{2} \)
67 \( 1 - 3.72e22iT - 4.48e45T^{2} \)
71 \( 1 + 3.96e22T + 1.91e46T^{2} \)
73 \( 1 + 4.27e22iT - 3.82e46T^{2} \)
79 \( 1 + 9.25e23T + 2.75e47T^{2} \)
83 \( 1 - 9.81e23iT - 9.48e47T^{2} \)
89 \( 1 + 1.32e24T + 5.42e48T^{2} \)
97 \( 1 + 3.52e24iT - 4.66e49T^{2} \)
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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

−16.84839771937259359384045808029, −15.65934526390299624033032350466, −15.02124718859463909990262433910, −12.99430682202910756758082315163, −10.45663643022194299053926278401, −8.333682474705703697503698209486, −7.26171949338929456172710835506, −5.30514362374070209649339681703, −3.89654453779974703423830544386, −0.16404146604136142879683851644, 1.35636367444452129080985308098, 2.85957060914441028818374333419, 4.46869675513449706230258130013, 7.42886827050961339730153498102, 9.456477817840484998602382989250, 11.26770728761369672012361070548, 12.20060048013250790684964960594, 13.52415005561806335899965574495, 15.81732705709704185370820889723, 18.45316108794090170591025077601

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