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Properties

Label 2-105-15.14-c2-0-2
Degree $2$
Conductor $105$
Sign $-0.886 - 0.463i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.50·2-s + (−2.08 + 2.15i)3-s − 1.72·4-s + (−4.79 + 1.41i)5-s + (−3.14 + 3.24i)6-s + 2.64i·7-s − 8.63·8-s + (−0.288 − 8.99i)9-s + (−7.22 + 2.13i)10-s + 0.491i·11-s + (3.60 − 3.72i)12-s + 21.9i·13-s + 3.98i·14-s + (6.95 − 13.2i)15-s − 6.11·16-s + 4.43·17-s + ⋯
L(s)  = 1  + 0.753·2-s + (−0.695 + 0.718i)3-s − 0.431·4-s + (−0.958 + 0.283i)5-s + (−0.524 + 0.541i)6-s + 0.377i·7-s − 1.07·8-s + (−0.0321 − 0.999i)9-s + (−0.722 + 0.213i)10-s + 0.0446i·11-s + (0.300 − 0.310i)12-s + 1.68i·13-s + 0.284i·14-s + (0.463 − 0.886i)15-s − 0.381·16-s + 0.260·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.886 - 0.463i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.886 - 0.463i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.159395 + 0.648790i\)
\(L(\frac12)\) \(\approx\) \(0.159395 + 0.648790i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.08 - 2.15i)T \)
5 \( 1 + (4.79 - 1.41i)T \)
7 \( 1 - 2.64iT \)
good2 \( 1 - 1.50T + 4T^{2} \)
11 \( 1 - 0.491iT - 121T^{2} \)
13 \( 1 - 21.9iT - 169T^{2} \)
17 \( 1 - 4.43T + 289T^{2} \)
19 \( 1 - 14.4T + 361T^{2} \)
23 \( 1 + 19.9T + 529T^{2} \)
29 \( 1 - 35.6iT - 841T^{2} \)
31 \( 1 + 45.5T + 961T^{2} \)
37 \( 1 + 58.9iT - 1.36e3T^{2} \)
41 \( 1 - 45.7iT - 1.68e3T^{2} \)
43 \( 1 - 2.94iT - 1.84e3T^{2} \)
47 \( 1 + 41.5T + 2.20e3T^{2} \)
53 \( 1 - 18.4T + 2.80e3T^{2} \)
59 \( 1 - 9.08iT - 3.48e3T^{2} \)
61 \( 1 - 46.3T + 3.72e3T^{2} \)
67 \( 1 - 92.9iT - 4.48e3T^{2} \)
71 \( 1 - 37.1iT - 5.04e3T^{2} \)
73 \( 1 + 4.57iT - 5.32e3T^{2} \)
79 \( 1 + 80.9T + 6.24e3T^{2} \)
83 \( 1 - 14.0T + 6.88e3T^{2} \)
89 \( 1 - 155. iT - 7.92e3T^{2} \)
97 \( 1 + 124. iT - 9.40e3T^{2} \)
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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

−14.30721599248308020919402269401, −12.69454013531909457265754249303, −11.86575203311258089924776125413, −11.19703388399283144385299497805, −9.672183611492709321450929843258, −8.756696584728712548785238271852, −6.97815030694815061116847401894, −5.62857946796845684327920926501, −4.45420228136607896505420761449, −3.54817719758390702713834180149, 0.43459637001000193295654526509, 3.45414372476262000931837399621, 4.90047884720251738114473859162, 5.88203019786772423452411658480, 7.49045116280383043460131199961, 8.323276535041873070258121752185, 10.08404337450807163110656384638, 11.39234181967562530548225719254, 12.26536905232749634584300170603, 12.97656629448821278673651060361

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