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Properties

Label 2-105-105.59-c1-0-8
Degree $2$
Conductor $105$
Sign $0.842 - 0.538i$
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  + (0.757 + 1.31i)2-s + (1.24 − 1.20i)3-s + (−0.147 + 0.254i)4-s + (−2.20 + 0.376i)5-s + (2.52 + 0.722i)6-s + (0.0753 + 2.64i)7-s + 2.58·8-s + (0.102 − 2.99i)9-s + (−2.16 − 2.60i)10-s + (−1.86 − 1.07i)11-s + (0.123 + 0.494i)12-s − 3.48·13-s + (−3.41 + 2.10i)14-s + (−2.29 + 3.12i)15-s + (2.25 + 3.89i)16-s + (−3.09 − 1.78i)17-s + ⋯
L(s)  = 1  + (0.535 + 0.927i)2-s + (0.719 − 0.694i)3-s + (−0.0735 + 0.127i)4-s + (−0.985 + 0.168i)5-s + (1.02 + 0.294i)6-s + (0.0284 + 0.999i)7-s + 0.913·8-s + (0.0341 − 0.999i)9-s + (−0.684 − 0.824i)10-s + (−0.560 − 0.323i)11-s + (0.0356 + 0.142i)12-s − 0.965·13-s + (−0.911 + 0.561i)14-s + (−0.591 + 0.806i)15-s + (0.562 + 0.974i)16-s + (−0.751 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38075 + 0.403938i\)
\(L(\frac12)\) \(\approx\) \(1.38075 + 0.403938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.24 + 1.20i)T \)
5 \( 1 + (2.20 - 0.376i)T \)
7 \( 1 + (-0.0753 - 2.64i)T \)
good2 \( 1 + (-0.757 - 1.31i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.86 + 1.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + (3.09 + 1.78i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.05 - 0.611i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.757 + 1.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-2.75 - 1.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.75 + 3.90i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 2.99iT - 43T^{2} \)
47 \( 1 + (-5.28 + 3.05i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.61 - 9.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.08 + 1.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.94 - 1.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.93 + 5.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (3.42 - 5.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.941 - 1.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.10iT - 83T^{2} \)
89 \( 1 + (0.889 + 1.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.32T + 97T^{2} \)
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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

−14.19996557753244436790595339961, −12.94388349515885161745585484429, −12.15429047766255584858753997772, −10.87492849733600299489742406490, −9.125009339463468202939900673790, −7.990702447025926801990227242154, −7.23627338470104153666493784431, −6.06461226252646706323529219743, −4.57725579842541001299472135157, −2.67101587482838326273453714648, 2.62401308499100439557257920338, 4.06552960414642397149645808914, 4.60979048643284914603997842929, 7.38724667306247141909888360744, 8.071398926644802146938712001860, 9.732112198534577219521407370205, 10.67789851358849167599814139371, 11.47149830395532256173577349485, 12.72532310642147688606762497205, 13.47181240554719227778606070154

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