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Properties

Label 2-10-5.4-c25-0-0
Degree $2$
Conductor $10$
Sign $0.904 + 0.426i$
Analytic cond. $39.5996$
Root an. cond. $6.29282$
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  + 4.09e3i·2-s + 1.47e6i·3-s − 1.67e7·4-s + (−2.33e8 + 4.93e8i)5-s − 6.04e9·6-s + 3.57e9i·7-s − 6.87e10i·8-s − 1.32e12·9-s + (−2.02e12 − 9.54e11i)10-s + 1.08e12·11-s − 2.47e13i·12-s − 2.13e13i·13-s − 1.46e13·14-s + (−7.28e14 − 3.43e14i)15-s + 2.81e14·16-s − 3.79e15i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.60i·3-s − 0.5·4-s + (−0.426 + 0.904i)5-s − 1.13·6-s + 0.0977i·7-s − 0.353i·8-s − 1.56·9-s + (−0.639 − 0.301i)10-s + 0.104·11-s − 0.801i·12-s − 0.253i·13-s − 0.0691·14-s + (−1.44 − 0.683i)15-s + 0.250·16-s − 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(39.5996\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :25/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(13)\) \(\approx\) \(0.1987338539\)
\(L(\frac12)\) \(\approx\) \(0.1987338539\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4.09e3iT \)
5 \( 1 + (2.33e8 - 4.93e8i)T \)
good3 \( 1 - 1.47e6iT - 8.47e11T^{2} \)
7 \( 1 - 3.57e9iT - 1.34e21T^{2} \)
11 \( 1 - 1.08e12T + 1.08e26T^{2} \)
13 \( 1 + 2.13e13iT - 7.05e27T^{2} \)
17 \( 1 + 3.79e15iT - 5.77e30T^{2} \)
19 \( 1 + 6.90e15T + 9.30e31T^{2} \)
23 \( 1 - 1.56e17iT - 1.10e34T^{2} \)
29 \( 1 + 6.08e17T + 3.63e36T^{2} \)
31 \( 1 + 3.42e18T + 1.92e37T^{2} \)
37 \( 1 - 7.61e19iT - 1.60e39T^{2} \)
41 \( 1 - 2.17e20T + 2.08e40T^{2} \)
43 \( 1 + 4.70e20iT - 6.86e40T^{2} \)
47 \( 1 + 6.53e20iT - 6.34e41T^{2} \)
53 \( 1 - 4.54e21iT - 1.27e43T^{2} \)
59 \( 1 + 1.39e22T + 1.86e44T^{2} \)
61 \( 1 + 4.30e21T + 4.29e44T^{2} \)
67 \( 1 - 3.04e22iT - 4.48e45T^{2} \)
71 \( 1 + 4.08e22T + 1.91e46T^{2} \)
73 \( 1 - 1.14e23iT - 3.82e46T^{2} \)
79 \( 1 + 6.98e23T + 2.75e47T^{2} \)
83 \( 1 + 1.18e24iT - 9.48e47T^{2} \)
89 \( 1 + 2.79e24T + 5.42e48T^{2} \)
97 \( 1 + 1.06e25iT - 4.66e49T^{2} \)
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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

−15.75126498488881340268540368460, −15.16784546914249843234121700179, −13.95626610531396953048005213132, −11.49809175695108955782180545728, −10.23201603915913380141914517413, −9.059825167771839553584290959857, −7.32419313157186902251876938195, −5.56648591801955870853064559325, −4.22186438162608798252107489096, −3.05176750830666439302333647322, 0.06257221142964261473911151660, 1.16572365751063118778217214935, 2.17963865267897399736559715200, 4.15203695792543450065737006544, 6.09078266708011139052981178525, 7.77022337037619241065436686591, 8.841259617357928091293007898409, 11.04732161115711336910459590572, 12.61730107994844757576776100956, 12.77456234380959855085776628638

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