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Properties

Label 2-5-5.4-c19-0-2
Degree $2$
Conductor $5$
Sign $0.929 + 0.369i$
Analytic cond. $11.4408$
Root an. cond. $3.38243$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21e3i·2-s + 1.87e4i·3-s − 9.54e5·4-s + (4.05e6 + 1.61e6i)5-s + 2.28e7·6-s + 1.79e8i·7-s + 5.22e8i·8-s + 8.10e8·9-s + (1.96e9 − 4.93e9i)10-s − 3.61e9·11-s − 1.78e10i·12-s − 7.48e9i·13-s + 2.18e11·14-s + (−3.03e10 + 7.60e10i)15-s + 1.35e11·16-s + 2.32e11i·17-s + ⋯
L(s)  = 1  − 1.67i·2-s + 0.550i·3-s − 1.81·4-s + (0.929 + 0.369i)5-s + 0.923·6-s + 1.68i·7-s + 1.37i·8-s + 0.697·9-s + (0.621 − 1.56i)10-s − 0.462·11-s − 1.00i·12-s − 0.195i·13-s + 2.82·14-s + (−0.203 + 0.511i)15-s + 0.491·16-s + 0.474i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(11.4408\)
Root analytic conductor: \(3.38243\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :19/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.80790 - 0.346719i\)
\(L(\frac12)\) \(\approx\) \(1.80790 - 0.346719i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.05e6 - 1.61e6i)T \)
good2 \( 1 + 1.21e3iT - 5.24e5T^{2} \)
3 \( 1 - 1.87e4iT - 1.16e9T^{2} \)
7 \( 1 - 1.79e8iT - 1.13e16T^{2} \)
11 \( 1 + 3.61e9T + 6.11e19T^{2} \)
13 \( 1 + 7.48e9iT - 1.46e21T^{2} \)
17 \( 1 - 2.32e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.12e12T + 1.97e24T^{2} \)
23 \( 1 - 6.69e12iT - 7.46e25T^{2} \)
29 \( 1 + 7.26e13T + 6.10e27T^{2} \)
31 \( 1 - 2.51e14T + 2.16e28T^{2} \)
37 \( 1 + 2.00e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.32e15T + 4.39e30T^{2} \)
43 \( 1 + 1.85e15iT - 1.08e31T^{2} \)
47 \( 1 + 9.40e15iT - 5.88e31T^{2} \)
53 \( 1 - 1.66e16iT - 5.77e32T^{2} \)
59 \( 1 + 1.59e15T + 4.42e33T^{2} \)
61 \( 1 + 4.33e16T + 8.34e33T^{2} \)
67 \( 1 - 2.54e17iT - 4.95e34T^{2} \)
71 \( 1 + 2.27e17T + 1.49e35T^{2} \)
73 \( 1 + 8.76e17iT - 2.53e35T^{2} \)
79 \( 1 - 5.76e17T + 1.13e36T^{2} \)
83 \( 1 + 5.61e17iT - 2.90e36T^{2} \)
89 \( 1 - 2.36e18T + 1.09e37T^{2} \)
97 \( 1 + 1.90e18iT - 5.60e37T^{2} \)
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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

−18.86244894693574377451691133918, −18.05154616297632255216266723450, −15.39907136537755027581787664478, −13.34615993691223862580192791601, −11.89831783572531198957355283866, −10.23544518592607272513778289597, −9.235877867419420404177398247918, −5.31572913527155533852192511861, −3.07478043977055208750766695212, −1.74882732092898545758802220120, 0.920789572771209375375736521027, 4.75903219157888522097274595731, 6.61734897906604316339147424740, 7.72271848101043549185123868727, 9.857281060033695054264620136204, 13.30362119482383243514890326804, 14.00664432499115017669385559795, 16.09965897907491999781915792917, 17.18760847733232842897333718694, 18.28954956664641263337829480722

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