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Properties

Label 2-10-5.4-c17-0-1
Degree $2$
Conductor $10$
Sign $-0.999 - 0.0263i$
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 + 1.58e4i·3-s − 6.55e4·4-s + (−2.29e4 + 8.73e5i)5-s + 4.05e6·6-s + 1.02e7i·7-s + 1.67e7i·8-s − 1.21e8·9-s + (2.23e8 + 5.88e6i)10-s − 2.30e8·11-s − 1.03e9i·12-s − 5.62e9i·13-s + 2.62e9·14-s + (−1.38e10 − 3.64e8i)15-s + 4.29e9·16-s + 4.57e10i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.39i·3-s − 0.5·4-s + (−0.0263 + 0.999i)5-s + 0.986·6-s + 0.673i·7-s + 0.353i·8-s − 0.944·9-s + (0.706 + 0.0186i)10-s − 0.323·11-s − 0.697i·12-s − 1.91i·13-s + 0.475·14-s + (−1.39 − 0.0366i)15-s + 0.250·16-s + 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0263i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.999 - 0.0263i$
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: \((2,\ 10,\ (\ :17/2),\ -0.999 - 0.0263i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.00997398 + 0.758079i\)
\(L(\frac12)\) \(\approx\) \(0.00997398 + 0.758079i\)
\(L(\frac{19}{2})\) 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 \( 1 + (2.29e4 - 8.73e5i)T \)
good3 \( 1 - 1.58e4iT - 1.29e8T^{2} \)
7 \( 1 - 1.02e7iT - 2.32e14T^{2} \)
11 \( 1 + 2.30e8T + 5.05e17T^{2} \)
13 \( 1 + 5.62e9iT - 8.65e18T^{2} \)
17 \( 1 - 4.57e10iT - 8.27e20T^{2} \)
19 \( 1 + 9.33e10T + 5.48e21T^{2} \)
23 \( 1 + 2.72e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.28e12T + 7.25e24T^{2} \)
31 \( 1 + 2.66e12T + 2.25e25T^{2} \)
37 \( 1 + 6.98e12iT - 4.56e26T^{2} \)
41 \( 1 + 1.41e13T + 2.61e27T^{2} \)
43 \( 1 + 7.85e13iT - 5.87e27T^{2} \)
47 \( 1 + 6.13e13iT - 2.66e28T^{2} \)
53 \( 1 - 4.49e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.79e15T + 1.27e30T^{2} \)
61 \( 1 - 1.23e15T + 2.24e30T^{2} \)
67 \( 1 - 6.13e15iT - 1.10e31T^{2} \)
71 \( 1 + 2.11e15T + 2.96e31T^{2} \)
73 \( 1 - 1.00e16iT - 4.74e31T^{2} \)
79 \( 1 + 9.45e15T + 1.81e32T^{2} \)
83 \( 1 - 3.02e16iT - 4.21e32T^{2} \)
89 \( 1 - 3.43e16T + 1.37e33T^{2} \)
97 \( 1 - 6.26e16iT - 5.95e33T^{2} \)
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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

−17.38113383106527190455429597931, −15.42556100087001007184776021768, −14.82502804213846429605822470958, −12.69004486843759661152081366846, −10.69827021795687625136344844685, −10.31249851023221282632483889979, −8.473158383416687392622638358881, −5.66985040921794420977329818557, −3.88175506143158772417055426535, −2.61302536940863061858922491186, 0.27930740461342307542230315967, 1.67453256327570854676197864111, 4.59656205742445001981506762839, 6.55199637204741266837098414583, 7.66746803348276493033704028272, 9.161397506662088638329481113790, 11.83107649075141733491751087922, 13.19165403266593447977187947595, 14.01590219348667832910500176735, 16.19492865077235502914623128212

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