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Properties

Label 2-10-5.4-c17-0-4
Degree $2$
Conductor $10$
Sign $0.627 + 0.778i$
Analytic cond. $18.3222$
Root an. cond. $4.28044$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 256i·2-s − 5.43e3i·3-s − 6.55e4·4-s + (−6.79e5 + 5.48e5i)5-s + 1.39e6·6-s + 7.62e6i·7-s − 1.67e7i·8-s + 9.95e7·9-s + (−1.40e8 − 1.74e8i)10-s − 8.00e8·11-s + 3.56e8i·12-s − 4.04e9i·13-s − 1.95e9·14-s + (2.98e9 + 3.69e9i)15-s + 4.29e9·16-s − 3.05e10i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.478i·3-s − 0.5·4-s + (−0.778 + 0.627i)5-s + 0.338·6-s + 0.500i·7-s − 0.353i·8-s + 0.771·9-s + (−0.443 − 0.550i)10-s − 1.12·11-s + 0.239i·12-s − 1.37i·13-s − 0.353·14-s + (0.300 + 0.372i)15-s + 0.250·16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(18.3222\)
Root analytic conductor: \(4.28044\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :17/2),\ 0.627 + 0.778i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.949360 - 0.453969i\)
\(L(\frac12)\) \(\approx\) \(0.949360 - 0.453969i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 256iT \)
5 \( 1 + (6.79e5 - 5.48e5i)T \)
good3 \( 1 + 5.43e3iT - 1.29e8T^{2} \)
7 \( 1 - 7.62e6iT - 2.32e14T^{2} \)
11 \( 1 + 8.00e8T + 5.05e17T^{2} \)
13 \( 1 + 4.04e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.05e10iT - 8.27e20T^{2} \)
19 \( 1 - 1.17e11T + 5.48e21T^{2} \)
23 \( 1 + 2.47e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.59e12T + 7.25e24T^{2} \)
31 \( 1 + 2.34e12T + 2.25e25T^{2} \)
37 \( 1 - 1.55e13iT - 4.56e26T^{2} \)
41 \( 1 + 7.85e13T + 2.61e27T^{2} \)
43 \( 1 + 4.97e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.34e14iT - 2.66e28T^{2} \)
53 \( 1 + 4.51e14iT - 2.05e29T^{2} \)
59 \( 1 + 4.74e14T + 1.27e30T^{2} \)
61 \( 1 + 2.35e15T + 2.24e30T^{2} \)
67 \( 1 - 2.78e15iT - 1.10e31T^{2} \)
71 \( 1 - 2.28e15T + 2.96e31T^{2} \)
73 \( 1 - 5.13e15iT - 4.74e31T^{2} \)
79 \( 1 - 1.83e16T + 1.81e32T^{2} \)
83 \( 1 + 2.16e16iT - 4.21e32T^{2} \)
89 \( 1 + 6.72e15T + 1.37e33T^{2} \)
97 \( 1 + 9.88e16iT - 5.95e33T^{2} \)
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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

−15.95685825563596401914087430213, −15.27141264241983699939424693404, −13.53929674538742264906438009243, −12.13352960738950341324498978414, −10.21697505890438122501996091288, −8.050346651373740781023835624398, −7.08516836578308420471120598972, −5.19044842520786329292743072429, −2.99687082525369721323533014953, −0.43813010996742229285315692593, 1.35668830672306573214905677983, 3.67315229098402979657601252390, 4.84086105601317868516634201920, 7.62798784661814710014465004539, 9.374636343599750197752108892404, 10.78595562475987782319720034915, 12.23211129233270850726332459076, 13.59690616162421417267492839158, 15.52334924194022583174031725263, 16.61536245032276493042324733089

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