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Properties

Label 2-10-5.2-c18-0-5
Degree $2$
Conductor $10$
Sign $0.465 + 0.884i$
Analytic cond. $20.5386$
Root an. cond. $4.53195$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−256 − 256i)2-s + (4.47e3 − 4.47e3i)3-s + 1.31e5i·4-s + (1.47e6 − 1.28e6i)5-s − 2.29e6·6-s + (−6.84e6 − 6.84e6i)7-s + (3.35e7 − 3.35e7i)8-s + 3.47e8i·9-s + (−7.05e8 − 4.88e7i)10-s + 4.51e9·11-s + (5.86e8 + 5.86e8i)12-s + (−9.92e9 + 9.92e9i)13-s + 3.50e9i·14-s + (8.53e8 − 1.23e10i)15-s − 1.71e10·16-s + (8.71e10 + 8.71e10i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.227 − 0.227i)3-s + 0.5i·4-s + (0.754 − 0.656i)5-s − 0.227·6-s + (−0.169 − 0.169i)7-s + (0.250 − 0.250i)8-s + 0.896i·9-s + (−0.705 − 0.0488i)10-s + 1.91·11-s + (0.113 + 0.113i)12-s + (−0.935 + 0.935i)13-s + 0.169i·14-s + (0.0222 − 0.320i)15-s − 0.250·16-s + (0.734 + 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.465 + 0.884i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ 0.465 + 0.884i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.66655 - 1.00619i\)
\(L(\frac12)\) \(\approx\) \(1.66655 - 1.00619i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (256 + 256i)T \)
5 \( 1 + (-1.47e6 + 1.28e6i)T \)
good3 \( 1 + (-4.47e3 + 4.47e3i)T - 3.87e8iT^{2} \)
7 \( 1 + (6.84e6 + 6.84e6i)T + 1.62e15iT^{2} \)
11 \( 1 - 4.51e9T + 5.55e18T^{2} \)
13 \( 1 + (9.92e9 - 9.92e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (-8.71e10 - 8.71e10i)T + 1.40e22iT^{2} \)
19 \( 1 + 5.76e11iT - 1.04e23T^{2} \)
23 \( 1 + (-1.51e12 + 1.51e12i)T - 3.24e24iT^{2} \)
29 \( 1 + 1.77e13iT - 2.10e26T^{2} \)
31 \( 1 - 1.04e13T + 6.99e26T^{2} \)
37 \( 1 + (-2.32e13 - 2.32e13i)T + 1.68e28iT^{2} \)
41 \( 1 - 9.57e13T + 1.07e29T^{2} \)
43 \( 1 + (3.63e14 - 3.63e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (-4.58e14 - 4.58e14i)T + 1.25e30iT^{2} \)
53 \( 1 + (-1.61e15 + 1.61e15i)T - 1.08e31iT^{2} \)
59 \( 1 + 2.05e15iT - 7.50e31T^{2} \)
61 \( 1 - 1.50e16T + 1.36e32T^{2} \)
67 \( 1 + (7.90e15 + 7.90e15i)T + 7.40e32iT^{2} \)
71 \( 1 - 1.03e16T + 2.10e33T^{2} \)
73 \( 1 + (7.59e16 - 7.59e16i)T - 3.46e33iT^{2} \)
79 \( 1 - 8.98e16iT - 1.43e34T^{2} \)
83 \( 1 + (2.16e16 - 2.16e16i)T - 3.49e34iT^{2} \)
89 \( 1 + 2.06e17iT - 1.22e35T^{2} \)
97 \( 1 + (3.83e17 + 3.83e17i)T + 5.77e35iT^{2} \)
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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

−16.73611291811665241619950990499, −14.34189456005076788294446057452, −13.06776038617392362123375225617, −11.61798530347989815432390368166, −9.791747401676619466992951321947, −8.718856028827354531712932589312, −6.78783198592864604795892990222, −4.51979065707134307629489944054, −2.28751239651698662368957315338, −1.01601820127703971604767765435, 1.23832883543070481471621529707, 3.33093579919455093308471489336, 5.78487808684855362433039306199, 7.09915940463975821820104710621, 9.165471348335427839975444664432, 10.05466698566369201291856398097, 12.07844002244932090938940854462, 14.29673731175918056918017817180, 14.92001786408258036289461147775, 16.79149132012657158797002161946

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