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Properties

Label 2-105-35.13-c3-0-6
Degree $2$
Conductor $105$
Sign $-0.716 - 0.697i$
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  + (−3.58 + 3.58i)2-s + (−2.12 + 2.12i)3-s − 17.7i·4-s + (10.0 − 4.93i)5-s − 15.2i·6-s + (2.02 + 18.4i)7-s + (34.9 + 34.9i)8-s − 8.99i·9-s + (−18.2 + 53.6i)10-s + 70.3·11-s + (37.6 + 37.6i)12-s + (−20.4 + 20.4i)13-s + (−73.2 − 58.7i)14-s + (−10.8 + 31.7i)15-s − 108.·16-s + (0.412 + 0.412i)17-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)2-s + (−0.408 + 0.408i)3-s − 2.21i·4-s + (0.897 − 0.441i)5-s − 1.03i·6-s + (0.109 + 0.994i)7-s + (1.54 + 1.54i)8-s − 0.333i·9-s + (−0.577 + 1.69i)10-s + 1.92·11-s + (0.905 + 0.905i)12-s + (−0.436 + 0.436i)13-s + (−1.39 − 1.12i)14-s + (−0.185 + 0.546i)15-s − 1.69·16-s + (0.00588 + 0.00588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.716 - 0.697i$
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: \((2,\ 105,\ (\ :3/2),\ -0.716 - 0.697i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.313799 + 0.772115i\)
\(L(\frac12)\) \(\approx\) \(0.313799 + 0.772115i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.0 + 4.93i)T \)
7 \( 1 + (-2.02 - 18.4i)T \)
good2 \( 1 + (3.58 - 3.58i)T - 8iT^{2} \)
11 \( 1 - 70.3T + 1.33e3T^{2} \)
13 \( 1 + (20.4 - 20.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (-0.412 - 0.412i)T + 4.91e3iT^{2} \)
19 \( 1 + 88.5T + 6.85e3T^{2} \)
23 \( 1 + (-37.1 - 37.1i)T + 1.21e4iT^{2} \)
29 \( 1 - 116. iT - 2.43e4T^{2} \)
31 \( 1 - 227. iT - 2.97e4T^{2} \)
37 \( 1 + (-83.7 + 83.7i)T - 5.06e4iT^{2} \)
41 \( 1 - 438. iT - 6.89e4T^{2} \)
43 \( 1 + (-199. - 199. i)T + 7.95e4iT^{2} \)
47 \( 1 + (72.8 + 72.8i)T + 1.03e5iT^{2} \)
53 \( 1 + (141. + 141. i)T + 1.48e5iT^{2} \)
59 \( 1 - 380.T + 2.05e5T^{2} \)
61 \( 1 + 349. iT - 2.26e5T^{2} \)
67 \( 1 + (-8.87 + 8.87i)T - 3.00e5iT^{2} \)
71 \( 1 - 860.T + 3.57e5T^{2} \)
73 \( 1 + (817. - 817. i)T - 3.89e5iT^{2} \)
79 \( 1 + 698. iT - 4.93e5T^{2} \)
83 \( 1 + (-191. + 191. i)T - 5.71e5iT^{2} \)
89 \( 1 - 412.T + 7.04e5T^{2} \)
97 \( 1 + (-267. - 267. i)T + 9.12e5iT^{2} \)
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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

−14.42281432165919353317762649673, −12.54681963617193208342598485493, −11.27611978431348834828356430084, −9.871317137450458533679540214509, −9.173053791524223364637791222847, −8.591508125320062714098766813074, −6.72877068689634425248557710739, −6.09058398622980943049020859963, −4.87479549686559991012795709697, −1.47609586379096566573630687870, 0.839444618627395176457240001020, 2.15375739167341225129557800954, 3.95923576870500180186403822271, 6.39600590602281939910575579943, 7.46898250428877997196783654430, 8.908621094180330333929627179275, 9.867801006998296606097712218355, 10.69127473304953022505019695628, 11.51275553540268879074715029353, 12.54645428023701065931257476965

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