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Properties

Label 2-105-105.62-c2-0-19
Degree $2$
Conductor $105$
Sign $0.943 + 0.332i$
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  + (1.88 − 1.88i)2-s + (1.39 + 2.65i)3-s − 3.12i·4-s + (4.96 + 0.551i)5-s + (7.64 + 2.36i)6-s + (−6.68 + 2.09i)7-s + (1.65 + 1.65i)8-s + (−5.08 + 7.42i)9-s + (10.4 − 8.33i)10-s − 17.9i·11-s + (8.28 − 4.36i)12-s + (−11.1 − 11.1i)13-s + (−8.66 + 16.5i)14-s + (5.48 + 13.9i)15-s + 18.7·16-s + (−0.666 − 0.666i)17-s + ⋯
L(s)  = 1  + (0.943 − 0.943i)2-s + (0.466 + 0.884i)3-s − 0.780i·4-s + (0.993 + 0.110i)5-s + (1.27 + 0.394i)6-s + (−0.954 + 0.298i)7-s + (0.207 + 0.207i)8-s + (−0.564 + 0.825i)9-s + (1.04 − 0.833i)10-s − 1.62i·11-s + (0.690 − 0.363i)12-s + (−0.856 − 0.856i)13-s + (−0.618 + 1.18i)14-s + (0.365 + 0.930i)15-s + 1.17·16-s + (−0.0392 − 0.0392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.943 + 0.332i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43871 - 0.417677i\)
\(L(\frac12)\) \(\approx\) \(2.43871 - 0.417677i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.39 - 2.65i)T \)
5 \( 1 + (-4.96 - 0.551i)T \)
7 \( 1 + (6.68 - 2.09i)T \)
good2 \( 1 + (-1.88 + 1.88i)T - 4iT^{2} \)
11 \( 1 + 17.9iT - 121T^{2} \)
13 \( 1 + (11.1 + 11.1i)T + 169iT^{2} \)
17 \( 1 + (0.666 + 0.666i)T + 289iT^{2} \)
19 \( 1 + 10.8T + 361T^{2} \)
23 \( 1 + (2.20 + 2.20i)T + 529iT^{2} \)
29 \( 1 + 22.9T + 841T^{2} \)
31 \( 1 - 26.1iT - 961T^{2} \)
37 \( 1 + (-41.6 - 41.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 6.85T + 1.68e3T^{2} \)
43 \( 1 + (37.6 - 37.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-5.55 - 5.55i)T + 2.20e3iT^{2} \)
53 \( 1 + (-32.4 - 32.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 99.8iT - 3.48e3T^{2} \)
61 \( 1 + 44.6iT - 3.72e3T^{2} \)
67 \( 1 + (18.0 + 18.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 6.35iT - 5.04e3T^{2} \)
73 \( 1 + (-55.3 - 55.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 59.7iT - 6.24e3T^{2} \)
83 \( 1 + (42.2 - 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 58.4iT - 7.92e3T^{2} \)
97 \( 1 + (-11.1 + 11.1i)T - 9.40e3iT^{2} \)
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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

−13.40618340261720967834842859785, −12.70969508149044740099615100711, −11.25861231352019219893968796874, −10.39486362983734112700967236247, −9.554723692498171105676395398686, −8.301201324865412502884822647545, −6.05035597700738105494392308708, −5.06798484680185848754542719122, −3.39480324309121579046911796854, −2.63314544263950430654729356485, 2.17363320427576425980906832399, 4.24073440840535049513152736554, 5.78206246639229772485037241059, 6.81868260740616139511376568149, 7.37199947910888949159404130991, 9.262182850050579757645053785322, 10.04793609039786176030195721737, 12.22068350665816401782640357929, 12.96256679105407231250171644214, 13.54039067706537473464990707374

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