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Properties

Label 2-105-35.13-c3-0-5
Degree $2$
Conductor $105$
Sign $-0.788 - 0.614i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.817 + 0.817i)2-s + (−2.12 + 2.12i)3-s + 6.66i·4-s + (8.83 + 6.85i)5-s − 3.46i·6-s + (18.0 + 4.27i)7-s + (−11.9 − 11.9i)8-s − 8.99i·9-s + (−12.8 + 1.61i)10-s − 14.7·11-s + (−14.1 − 14.1i)12-s + (−44.4 + 44.4i)13-s + (−18.2 + 11.2i)14-s + (−33.2 + 4.18i)15-s − 33.7·16-s + (47.4 + 47.4i)17-s + ⋯
L(s)  = 1  + (−0.289 + 0.289i)2-s + (−0.408 + 0.408i)3-s + 0.832i·4-s + (0.789 + 0.613i)5-s − 0.235i·6-s + (0.972 + 0.231i)7-s + (−0.529 − 0.529i)8-s − 0.333i·9-s + (−0.405 + 0.0509i)10-s − 0.404·11-s + (−0.340 − 0.340i)12-s + (−0.947 + 0.947i)13-s + (−0.347 + 0.214i)14-s + (−0.572 + 0.0720i)15-s − 0.526·16-s + (0.677 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.788 - 0.614i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.788 - 0.614i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.396603 + 1.15482i\)
\(L(\frac12)\) \(\approx\) \(0.396603 + 1.15482i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-8.83 - 6.85i)T \)
7 \( 1 + (-18.0 - 4.27i)T \)
good2 \( 1 + (0.817 - 0.817i)T - 8iT^{2} \)
11 \( 1 + 14.7T + 1.33e3T^{2} \)
13 \( 1 + (44.4 - 44.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (-47.4 - 47.4i)T + 4.91e3iT^{2} \)
19 \( 1 + 9.66T + 6.85e3T^{2} \)
23 \( 1 + (59.3 + 59.3i)T + 1.21e4iT^{2} \)
29 \( 1 + 228. iT - 2.43e4T^{2} \)
31 \( 1 - 10.3iT - 2.97e4T^{2} \)
37 \( 1 + (198. - 198. i)T - 5.06e4iT^{2} \)
41 \( 1 + 345. iT - 6.89e4T^{2} \)
43 \( 1 + (-333. - 333. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-152. - 152. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-348. - 348. i)T + 1.48e5iT^{2} \)
59 \( 1 - 593.T + 2.05e5T^{2} \)
61 \( 1 - 328. iT - 2.26e5T^{2} \)
67 \( 1 + (-626. + 626. i)T - 3.00e5iT^{2} \)
71 \( 1 - 360.T + 3.57e5T^{2} \)
73 \( 1 + (504. - 504. i)T - 3.89e5iT^{2} \)
79 \( 1 + 159. iT - 4.93e5T^{2} \)
83 \( 1 + (413. - 413. i)T - 5.71e5iT^{2} \)
89 \( 1 + 707.T + 7.04e5T^{2} \)
97 \( 1 + (-604. - 604. i)T + 9.12e5iT^{2} \)
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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

−13.86181905100837075125836954464, −12.45314656590397356361666249030, −11.61691517031899742270334998197, −10.43974484829310461811725823035, −9.413943728310460558121906272723, −8.177315653286973580618822065305, −7.04074180414354509424337177881, −5.75088833968029013401256285515, −4.26295022264662875804598271628, −2.38873573627892346033636334845, 0.797068183864384004736707322461, 2.15110360174920995511092283345, 5.11077315177346910841526064401, 5.54167844825389022468667001378, 7.28584460349292965934541208323, 8.590627434506418106623713049408, 9.884776521724131380019997332479, 10.59206457499809948463194810786, 11.76162570982327738992929349946, 12.78065713318376207043035621725

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