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Properties

Label 2-105-35.33-c1-0-4
Degree $2$
Conductor $105$
Sign $0.945 + 0.325i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.105 + 0.394i)2-s + (−0.965 − 0.258i)3-s + (1.58 − 0.916i)4-s + (−0.699 − 2.12i)5-s − 0.408i·6-s + (2.57 + 0.605i)7-s + (1.10 + 1.10i)8-s + (0.866 + 0.499i)9-s + (0.763 − 0.500i)10-s + (−0.463 − 0.803i)11-s + (−1.77 + 0.474i)12-s + (−4.08 + 4.08i)13-s + (0.0332 + 1.08i)14-s + (0.125 + 2.23i)15-s + (1.51 − 2.62i)16-s + (−0.192 + 0.719i)17-s + ⋯
L(s)  = 1  + (0.0747 + 0.278i)2-s + (−0.557 − 0.149i)3-s + (0.793 − 0.458i)4-s + (−0.312 − 0.949i)5-s − 0.166i·6-s + (0.973 + 0.228i)7-s + (0.391 + 0.391i)8-s + (0.288 + 0.166i)9-s + (0.241 − 0.158i)10-s + (−0.139 − 0.242i)11-s + (−0.511 + 0.136i)12-s + (−1.13 + 1.13i)13-s + (0.00889 + 0.288i)14-s + (0.0324 + 0.576i)15-s + (0.378 − 0.655i)16-s + (−0.0467 + 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04881 - 0.175465i\)
\(L(\frac12)\) \(\approx\) \(1.04881 - 0.175465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.699 + 2.12i)T \)
7 \( 1 + (-2.57 - 0.605i)T \)
good2 \( 1 + (-0.105 - 0.394i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.463 + 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.08 - 4.08i)T - 13iT^{2} \)
17 \( 1 + (0.192 - 0.719i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.21 + 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.00 - 1.34i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 - 0.607i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.472 + 1.76i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.97iT - 41T^{2} \)
43 \( 1 + (0.781 + 0.781i)T + 43iT^{2} \)
47 \( 1 + (-10.0 + 2.70i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.72 + 6.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.72 + 2.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 + 2.80i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (-4.02 - 1.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.02 + 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.91 + 5.91i)T - 83iT^{2} \)
89 \( 1 + (7.78 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−13.87523598644091604492093435617, −12.32052115335750029461883530587, −11.72626505099733004342308725703, −10.82484613979576595334958915461, −9.390229962414136257392080040879, −8.011118524281105527875348150494, −6.98073125690107231508412149544, −5.51869551043764765272805386722, −4.66921578842305582329254071776, −1.78870627076827331208754799056, 2.49704968775688815186806713699, 4.16057781502314726825138973456, 5.84569482367469925440705278237, 7.33869465655432259756306614165, 7.85985164149476952480488772344, 10.14089751220947702203156683449, 10.70050641135917270555872662636, 11.77382437243928045331993805335, 12.30667715271585467297392965511, 13.87326604582351580386364732480

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