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Properties

Label 2-105-5.2-c2-0-5
Degree $2$
Conductor $105$
Sign $0.963 + 0.267i$
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  + (0.675 + 0.675i)2-s + (1.22 − 1.22i)3-s − 3.08i·4-s + (3.39 + 3.67i)5-s + 1.65·6-s + (−1.87 − 1.87i)7-s + (4.78 − 4.78i)8-s − 2.99i·9-s + (−0.186 + 4.77i)10-s + 7.59·11-s + (−3.78 − 3.78i)12-s + (1.12 − 1.12i)13-s − 2.52i·14-s + (8.65 + 0.337i)15-s − 5.88·16-s + (−3.43 − 3.43i)17-s + ⋯
L(s)  = 1  + (0.337 + 0.337i)2-s + (0.408 − 0.408i)3-s − 0.771i·4-s + (0.678 + 0.734i)5-s + 0.275·6-s + (−0.267 − 0.267i)7-s + (0.598 − 0.598i)8-s − 0.333i·9-s + (−0.0186 + 0.477i)10-s + 0.690·11-s + (−0.315 − 0.315i)12-s + (0.0863 − 0.0863i)13-s − 0.180i·14-s + (0.576 + 0.0225i)15-s − 0.367·16-s + (−0.202 − 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.963 + 0.267i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.963 + 0.267i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90944 - 0.260183i\)
\(L(\frac12)\) \(\approx\) \(1.90944 - 0.260183i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-3.39 - 3.67i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-0.675 - 0.675i)T + 4iT^{2} \)
11 \( 1 - 7.59T + 121T^{2} \)
13 \( 1 + (-1.12 + 1.12i)T - 169iT^{2} \)
17 \( 1 + (3.43 + 3.43i)T + 289iT^{2} \)
19 \( 1 - 26.3iT - 361T^{2} \)
23 \( 1 + (24.2 - 24.2i)T - 529iT^{2} \)
29 \( 1 + 22.3iT - 841T^{2} \)
31 \( 1 + 18.3T + 961T^{2} \)
37 \( 1 + (34.4 + 34.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 37.4T + 1.68e3T^{2} \)
43 \( 1 + (55.1 - 55.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-40.8 - 40.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (9.39 - 9.39i)T - 2.80e3iT^{2} \)
59 \( 1 + 49.7iT - 3.48e3T^{2} \)
61 \( 1 - 88.3T + 3.72e3T^{2} \)
67 \( 1 + (40.5 + 40.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 136.T + 5.04e3T^{2} \)
73 \( 1 + (-5.85 + 5.85i)T - 5.32e3iT^{2} \)
79 \( 1 + 66.4iT - 6.24e3T^{2} \)
83 \( 1 + (34.8 - 34.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 2.03iT - 7.92e3T^{2} \)
97 \( 1 + (58.4 + 58.4i)T + 9.40e3iT^{2} \)
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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

−13.94433626073824245788961924982, −12.76048021957725830151903223262, −11.30319421366029411702272081001, −10.10322795155976706823525379207, −9.409251278266254373468500545383, −7.64293078726937443585621505790, −6.52016089026607719349997592325, −5.72570194445150865673831818929, −3.77433819928230511262116627695, −1.77388266408271202743049270835, 2.29913221563469416565161512649, 3.89059188715993919794225499585, 5.09368617839340368172185134598, 6.78789255955853411316682729201, 8.483821424593878937681249172969, 9.051485342412645457670530279202, 10.35855406929580562388784635781, 11.70028463183214826510525415212, 12.62543839278077829348834992092, 13.47016623969460810431573213901

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