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Properties

Label 2-10-5.3-c16-0-4
Degree $2$
Conductor $10$
Sign $0.634 + 0.772i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (128 − 128i)2-s + (3.52e3 + 3.52e3i)3-s − 3.27e4i·4-s + (−3.50e5 + 1.72e5i)5-s + 9.01e5·6-s + (5.84e6 − 5.84e6i)7-s + (−4.19e6 − 4.19e6i)8-s − 1.82e7i·9-s + (−2.28e7 + 6.69e7i)10-s + 2.70e8·11-s + (1.15e8 − 1.15e8i)12-s + (6.31e8 + 6.31e8i)13-s − 1.49e9i·14-s + (−1.84e9 − 6.29e8i)15-s − 1.07e9·16-s + (6.03e9 − 6.03e9i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.536 + 0.536i)3-s − 0.5i·4-s + (−0.897 + 0.440i)5-s + 0.536·6-s + (1.01 − 1.01i)7-s + (−0.250 − 0.250i)8-s − 0.423i·9-s + (−0.228 + 0.669i)10-s + 1.26·11-s + (0.268 − 0.268i)12-s + (0.774 + 0.774i)13-s − 1.01i·14-s + (−0.718 − 0.245i)15-s − 0.250·16-s + (0.865 − 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ 0.634 + 0.772i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(2.63536 - 1.24521i\)
\(L(\frac12)\) \(\approx\) \(2.63536 - 1.24521i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-128 + 128i)T \)
5 \( 1 + (3.50e5 - 1.72e5i)T \)
good3 \( 1 + (-3.52e3 - 3.52e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-5.84e6 + 5.84e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 2.70e8T + 4.59e16T^{2} \)
13 \( 1 + (-6.31e8 - 6.31e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-6.03e9 + 6.03e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 1.33e10iT - 2.88e20T^{2} \)
23 \( 1 + (3.14e10 + 3.14e10i)T + 6.13e21iT^{2} \)
29 \( 1 - 8.33e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.45e12T + 7.27e23T^{2} \)
37 \( 1 + (-2.63e11 + 2.63e11i)T - 1.23e25iT^{2} \)
41 \( 1 + 2.13e12T + 6.37e25T^{2} \)
43 \( 1 + (-3.06e12 - 3.06e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (9.62e11 - 9.62e11i)T - 5.66e26iT^{2} \)
53 \( 1 + (-5.01e13 - 5.01e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 6.71e13iT - 2.15e28T^{2} \)
61 \( 1 + 4.32e13T + 3.67e28T^{2} \)
67 \( 1 + (1.98e14 - 1.98e14i)T - 1.64e29iT^{2} \)
71 \( 1 - 4.57e14T + 4.16e29T^{2} \)
73 \( 1 + (8.04e13 + 8.04e13i)T + 6.50e29iT^{2} \)
79 \( 1 + 1.23e15iT - 2.30e30T^{2} \)
83 \( 1 + (2.68e15 + 2.68e15i)T + 5.07e30iT^{2} \)
89 \( 1 - 4.97e15iT - 1.54e31T^{2} \)
97 \( 1 + (1.04e16 - 1.04e16i)T - 6.14e31iT^{2} \)
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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

−16.34546536418407616873421816495, −14.68382966049333904469202663522, −14.14570451595004276745906879043, −11.87654594121685799284218480289, −10.84412854306847557720170840650, −9.006183267071686821167669529462, −7.05084524931561812413408692679, −4.35602074110247673144498508129, −3.48570393828321549979865431849, −1.10899954393508874806138082486, 1.62857021480821082556675415912, 3.78475110100501149174631101031, 5.61741834920062036027091823232, 7.80536342000951861824849809953, 8.547789943387153564659109737967, 11.53796512086773731960019775447, 12.67284412430918359790190223222, 14.31089248056852846923983696584, 15.30323394348001322859678562767, 16.81487114800324441855341034459

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