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Properties

Label 2-105-7.4-c3-0-8
Degree $2$
Conductor $105$
Sign $0.620 - 0.784i$
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  + (1.70 + 2.95i)2-s + (1.5 − 2.59i)3-s + (−1.82 + 3.16i)4-s + (2.5 + 4.33i)5-s + 10.2·6-s + (16.7 − 7.91i)7-s + 14.8·8-s + (−4.5 − 7.79i)9-s + (−8.53 + 14.7i)10-s + (−18.8 + 32.6i)11-s + (5.48 + 9.50i)12-s + 43.7·13-s + (51.9 + 35.9i)14-s + 15.0·15-s + (39.9 + 69.1i)16-s + (−19.0 + 32.9i)17-s + ⋯
L(s)  = 1  + (0.603 + 1.04i)2-s + (0.288 − 0.499i)3-s + (−0.228 + 0.395i)4-s + (0.223 + 0.387i)5-s + 0.696·6-s + (0.904 − 0.427i)7-s + 0.655·8-s + (−0.166 − 0.288i)9-s + (−0.269 + 0.467i)10-s + (−0.517 + 0.895i)11-s + (0.131 + 0.228i)12-s + 0.933·13-s + (0.992 + 0.687i)14-s + 0.258·15-s + (0.624 + 1.08i)16-s + (−0.271 + 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.620 - 0.784i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.620 - 0.784i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.41307 + 1.16854i\)
\(L(\frac12)\) \(\approx\) \(2.41307 + 1.16854i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-16.7 + 7.91i)T \)
good2 \( 1 + (-1.70 - 2.95i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (18.8 - 32.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 43.7T + 2.19e3T^{2} \)
17 \( 1 + (19.0 - 32.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.7 + 86.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (1.38 + 2.39i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 + (-8.95 + 15.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (161. + 279. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 248.T + 6.89e4T^{2} \)
43 \( 1 + 474.T + 7.95e4T^{2} \)
47 \( 1 + (31.8 + 55.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-131. + 227. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-166. + 288. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-140. - 243. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (443. - 767. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 951.T + 3.57e5T^{2} \)
73 \( 1 + (406. - 703. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (260. + 451. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 339.T + 5.71e5T^{2} \)
89 \( 1 + (-688. - 1.19e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 194.T + 9.12e5T^{2} \)
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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

−13.52056845595488813985747570775, −12.99685444142784289055684007017, −11.29370007718450762730662245267, −10.38759255251139184910832673543, −8.633073251040782659702704994703, −7.52760226999496457116757489423, −6.76652282662650140758804102738, −5.48684992396402313529398169899, −4.17163824679879880479158155209, −1.87841712334366198302202168712, 1.73164909618390737586093779820, 3.26434154893318483568562906786, 4.56080314002054701644177843834, 5.71708238182665714472825365918, 7.942505448421718532537440712873, 8.836571760796948562540594622901, 10.31547553787083144823136291022, 11.12743934039804700073497024327, 11.95413610657021967888257219964, 13.19394136325701263195743225683

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