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Properties

Label 2-10-5.4-c15-0-5
Degree $2$
Conductor $10$
Sign $0.966 + 0.255i$
Analytic cond. $14.2693$
Root an. cond. $3.77747$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128i·2-s + 3.70e3i·3-s − 1.63e4·4-s + (4.45e4 − 1.68e5i)5-s − 4.73e5·6-s − 2.91e6i·7-s − 2.09e6i·8-s + 6.38e5·9-s + (2.16e7 + 5.70e6i)10-s − 3.54e7·11-s − 6.06e7i·12-s − 3.71e7i·13-s + 3.73e8·14-s + (6.25e8 + 1.65e8i)15-s + 2.68e8·16-s − 5.94e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.977i·3-s − 0.5·4-s + (0.255 − 0.966i)5-s − 0.691·6-s − 1.33i·7-s − 0.353i·8-s + 0.0445·9-s + (0.683 + 0.180i)10-s − 0.548·11-s − 0.488i·12-s − 0.164i·13-s + 0.946·14-s + (0.945 + 0.249i)15-s + 0.250·16-s − 0.351i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(14.2693\)
Root analytic conductor: \(3.77747\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :15/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.53835 - 0.199653i\)
\(L(\frac12)\) \(\approx\) \(1.53835 - 0.199653i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 128iT \)
5 \( 1 + (-4.45e4 + 1.68e5i)T \)
good3 \( 1 - 3.70e3iT - 1.43e7T^{2} \)
7 \( 1 + 2.91e6iT - 4.74e12T^{2} \)
11 \( 1 + 3.54e7T + 4.17e15T^{2} \)
13 \( 1 + 3.71e7iT - 5.11e16T^{2} \)
17 \( 1 + 5.94e8iT - 2.86e18T^{2} \)
19 \( 1 - 5.75e9T + 1.51e19T^{2} \)
23 \( 1 + 2.28e10iT - 2.66e20T^{2} \)
29 \( 1 - 3.52e10T + 8.62e21T^{2} \)
31 \( 1 + 2.54e11T + 2.34e22T^{2} \)
37 \( 1 + 6.86e11iT - 3.33e23T^{2} \)
41 \( 1 - 2.30e12T + 1.55e24T^{2} \)
43 \( 1 + 9.79e11iT - 3.17e24T^{2} \)
47 \( 1 + 9.60e11iT - 1.20e25T^{2} \)
53 \( 1 + 7.41e12iT - 7.31e25T^{2} \)
59 \( 1 + 2.16e13T + 3.65e26T^{2} \)
61 \( 1 - 1.43e13T + 6.02e26T^{2} \)
67 \( 1 - 7.67e13iT - 2.46e27T^{2} \)
71 \( 1 + 4.18e13T + 5.87e27T^{2} \)
73 \( 1 + 9.53e13iT - 8.90e27T^{2} \)
79 \( 1 - 2.11e14T + 2.91e28T^{2} \)
83 \( 1 + 2.08e13iT - 6.11e28T^{2} \)
89 \( 1 + 7.21e14T + 1.74e29T^{2} \)
97 \( 1 - 1.18e15iT - 6.33e29T^{2} \)
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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

−16.52269478908631086724616390361, −16.02884305326901710856872419587, −14.25460815605163608928164995175, −12.92996112864135119939669330028, −10.49968760297905608141748717331, −9.283294958367445781589384577223, −7.48604620308123061459809313284, −5.21600469100104476892462502824, −4.03574423703267638464773305777, −0.67647026831961341767207590400, 1.66040905240296801128649671614, 2.92796108284191867752442929231, 5.76426673669639311184064340316, 7.57391880056435050947174171997, 9.555258086705046630486381076942, 11.34783033197026585721441100426, 12.57110189235445973759566053175, 13.88153705432347951804238467895, 15.46437544422961161032901965882, 17.99391124696083637149889068010

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