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Properties

Label 2-105-7.5-c2-0-0
Degree $2$
Conductor $105$
Sign $-0.582 - 0.813i$
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  + (−0.336 + 0.583i)2-s + (−1.5 + 0.866i)3-s + (1.77 + 3.07i)4-s + (1.93 + 1.11i)5-s − 1.16i·6-s + (−6.82 + 1.55i)7-s − 5.08·8-s + (1.5 − 2.59i)9-s + (−1.30 + 0.752i)10-s + (0.0223 + 0.0387i)11-s + (−5.31 − 3.07i)12-s + 23.0i·13-s + (1.38 − 4.50i)14-s − 3.87·15-s + (−5.38 + 9.32i)16-s + (−8.16 + 4.71i)17-s + ⋯
L(s)  = 1  + (−0.168 + 0.291i)2-s + (−0.5 + 0.288i)3-s + (0.443 + 0.767i)4-s + (0.387 + 0.223i)5-s − 0.194i·6-s + (−0.974 + 0.222i)7-s − 0.635·8-s + (0.166 − 0.288i)9-s + (−0.130 + 0.0752i)10-s + (0.00203 + 0.00352i)11-s + (−0.443 − 0.255i)12-s + 1.76i·13-s + (0.0992 − 0.321i)14-s − 0.258·15-s + (−0.336 + 0.582i)16-s + (−0.480 + 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.582 - 0.813i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.582 - 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.452897 + 0.881111i\)
\(L(\frac12)\) \(\approx\) \(0.452897 + 0.881111i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (6.82 - 1.55i)T \)
good2 \( 1 + (0.336 - 0.583i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-0.0223 - 0.0387i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 23.0iT - 169T^{2} \)
17 \( 1 + (8.16 - 4.71i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.991 - 0.572i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-22.1 + 38.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 53.0T + 841T^{2} \)
31 \( 1 + (-19.5 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (21.1 - 36.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 38.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.5T + 1.84e3T^{2} \)
47 \( 1 + (23.5 + 13.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (9.49 + 16.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (4.21 - 2.43i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.6 + 19.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-3.50 - 6.06i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 46.8T + 5.04e3T^{2} \)
73 \( 1 + (-72.3 + 41.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (10.2 - 17.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 125. iT - 6.88e3T^{2} \)
89 \( 1 + (-40.4 - 23.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 3.11iT - 9.40e3T^{2} \)
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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

−13.84507572225060753713825768832, −12.65063709604533991501082533005, −11.87294970163043742152201354515, −10.76270634359486228078570194919, −9.512620473547530742316341924255, −8.554385060547114977990262329105, −6.67085253903701881664410209718, −6.48317050323161396664016920623, −4.40275203480249749543121459282, −2.72573715326276973194961801925, 0.833142826775340788140930803211, 2.89933062494391018806016300811, 5.27292227560566692531178341960, 6.18021160223615948083412274311, 7.34253148671810718466428833513, 9.115463375122662790652720564908, 10.17988342292801541553202273630, 10.82522314945814281282932898833, 12.14094765231524895269043659498, 13.02618650316160683018610898674

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