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Properties

Label 2-105-15.14-c2-0-1
Degree $2$
Conductor $105$
Sign $-0.286 - 0.958i$
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  − 0.505·2-s + (−0.985 − 2.83i)3-s − 3.74·4-s + (−0.221 + 4.99i)5-s + (0.498 + 1.43i)6-s + 2.64i·7-s + 3.91·8-s + (−7.05 + 5.58i)9-s + (0.111 − 2.52i)10-s + 13.3i·11-s + (3.68 + 10.6i)12-s + 3.27i·13-s − 1.33i·14-s + (14.3 − 4.29i)15-s + 12.9·16-s − 21.9·17-s + ⋯
L(s)  = 1  − 0.252·2-s + (−0.328 − 0.944i)3-s − 0.936·4-s + (−0.0442 + 0.999i)5-s + (0.0830 + 0.238i)6-s + 0.377i·7-s + 0.489·8-s + (−0.784 + 0.620i)9-s + (0.0111 − 0.252i)10-s + 1.21i·11-s + (0.307 + 0.884i)12-s + 0.251i·13-s − 0.0955i·14-s + (0.958 − 0.286i)15-s + 0.812·16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.286 - 0.958i$
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: \((2,\ 105,\ (\ :1),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.275017 + 0.369251i\)
\(L(\frac12)\) \(\approx\) \(0.275017 + 0.369251i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.985 + 2.83i)T \)
5 \( 1 + (0.221 - 4.99i)T \)
7 \( 1 - 2.64iT \)
good2 \( 1 + 0.505T + 4T^{2} \)
11 \( 1 - 13.3iT - 121T^{2} \)
13 \( 1 - 3.27iT - 169T^{2} \)
17 \( 1 + 21.9T + 289T^{2} \)
19 \( 1 + 4.36T + 361T^{2} \)
23 \( 1 + 30.4T + 529T^{2} \)
29 \( 1 + 4.90iT - 841T^{2} \)
31 \( 1 - 29.2T + 961T^{2} \)
37 \( 1 + 24.8iT - 1.36e3T^{2} \)
41 \( 1 + 13.6iT - 1.68e3T^{2} \)
43 \( 1 - 64.9iT - 1.84e3T^{2} \)
47 \( 1 - 75.3T + 2.20e3T^{2} \)
53 \( 1 + 67.1T + 2.80e3T^{2} \)
59 \( 1 - 61.2iT - 3.48e3T^{2} \)
61 \( 1 - 46.8T + 3.72e3T^{2} \)
67 \( 1 + 96.7iT - 4.48e3T^{2} \)
71 \( 1 - 108. iT - 5.04e3T^{2} \)
73 \( 1 - 50.7iT - 5.32e3T^{2} \)
79 \( 1 + 59.8T + 6.24e3T^{2} \)
83 \( 1 - 82.9T + 6.88e3T^{2} \)
89 \( 1 + 79.4iT - 7.92e3T^{2} \)
97 \( 1 - 103. iT - 9.40e3T^{2} \)
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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

−13.79893026823734244984508006171, −12.80981503240199270709093514163, −11.84584461048817500124761436927, −10.66289223584762338652517733957, −9.538848414032449694764207340605, −8.230099643089115577370495893913, −7.18052378330780440441144253733, −6.05657783597158104134491785651, −4.41178358890202168207773044153, −2.22070172351183365821967660036, 0.39036201497498985488699361698, 3.84831621209312931575065898774, 4.79536066097431311633621422779, 5.97746784473555640717851246747, 8.241878060906655583756388624835, 8.862029944956879167717781278872, 9.896584008417774732544333760827, 10.89882616876717903595154095310, 12.11611159285524388363146281472, 13.39173077981561831971523144439

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