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Properties

Label 2-105-7.4-c3-0-15
Degree $2$
Conductor $105$
Sign $0.0726 - 0.997i$
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  + (−2.05 − 3.56i)2-s + (1.5 − 2.59i)3-s + (−4.47 + 7.75i)4-s + (−2.5 − 4.33i)5-s − 12.3·6-s + (−9.40 + 15.9i)7-s + 3.91·8-s + (−4.5 − 7.79i)9-s + (−10.2 + 17.8i)10-s + (−7.99 + 13.8i)11-s + (13.4 + 23.2i)12-s − 79.5·13-s + (76.2 + 0.710i)14-s − 15.0·15-s + (27.7 + 48.0i)16-s + (34.7 − 60.1i)17-s + ⋯
L(s)  = 1  + (−0.727 − 1.26i)2-s + (0.288 − 0.499i)3-s + (−0.559 + 0.968i)4-s + (−0.223 − 0.387i)5-s − 0.840·6-s + (−0.508 + 0.861i)7-s + 0.173·8-s + (−0.166 − 0.288i)9-s + (−0.325 + 0.563i)10-s + (−0.219 + 0.379i)11-s + (0.322 + 0.559i)12-s − 1.69·13-s + (1.45 + 0.0135i)14-s − 0.258·15-s + (0.433 + 0.750i)16-s + (0.495 − 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.0726 - 0.997i$
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: \((2,\ 105,\ (\ :3/2),\ 0.0726 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0858136 + 0.0797919i\)
\(L(\frac12)\) \(\approx\) \(0.0858136 + 0.0797919i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.40 - 15.9i)T \)
good2 \( 1 + (2.05 + 3.56i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (7.99 - 13.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 79.5T + 2.19e3T^{2} \)
17 \( 1 + (-34.7 + 60.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-20.6 - 35.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-5.63 - 9.76i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 16.1T + 2.43e4T^{2} \)
31 \( 1 + (35.5 - 61.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (178. + 308. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 426.T + 6.89e4T^{2} \)
43 \( 1 + 346.T + 7.95e4T^{2} \)
47 \( 1 + (70.1 + 121. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (171. - 296. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-384. + 665. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-292. - 507. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-91.8 + 159. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 881.T + 3.57e5T^{2} \)
73 \( 1 + (-6.04 + 10.4i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-591. - 1.02e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 245.T + 5.71e5T^{2} \)
89 \( 1 + (443. + 768. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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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

−12.19874937333016193472815340606, −11.75248486748094378437838263057, −10.10003220843861140084186689615, −9.446966791936477267484752253620, −8.420097268064269522989707240095, −7.14495703438856396635824726532, −5.28234077353946342076693839884, −3.16907127361294445337060577146, −2.01192984238878648650310747289, −0.07501443247195315781504312035, 3.25423417943154128874450718004, 5.03286630948949791104915895202, 6.58073636724109021242190182884, 7.46777456395347482530361929023, 8.417204424384160986715036727529, 9.762181814168087974502016028849, 10.31107220069089390365159689461, 11.91422905096293564474753043004, 13.40404645926077079855346662392, 14.57637713531385871779521887437

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