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Properties

Label 2-105-21.20-c3-0-21
Degree $2$
Conductor $105$
Sign $0.911 + 0.411i$
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  + 1.80i·2-s + (1.60 − 4.94i)3-s + 4.75·4-s + 5·5-s + (8.90 + 2.89i)6-s + (2.01 − 18.4i)7-s + 22.9i·8-s + (−21.8 − 15.8i)9-s + 9.01i·10-s − 11.3i·11-s + (7.64 − 23.4i)12-s − 25.5i·13-s + (33.1 + 3.62i)14-s + (8.04 − 24.7i)15-s − 3.42·16-s + 88.9·17-s + ⋯
L(s)  = 1  + 0.637i·2-s + (0.309 − 0.950i)3-s + 0.593·4-s + 0.447·5-s + (0.605 + 0.197i)6-s + (0.108 − 0.994i)7-s + 1.01i·8-s + (−0.808 − 0.588i)9-s + 0.285i·10-s − 0.312i·11-s + (0.183 − 0.564i)12-s − 0.545i·13-s + (0.633 + 0.0692i)14-s + (0.138 − 0.425i)15-s − 0.0535·16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.09913 - 0.451339i\)
\(L(\frac12)\) \(\approx\) \(2.09913 - 0.451339i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 + 4.94i)T \)
5 \( 1 - 5T \)
7 \( 1 + (-2.01 + 18.4i)T \)
good2 \( 1 - 1.80iT - 8T^{2} \)
11 \( 1 + 11.3iT - 1.33e3T^{2} \)
13 \( 1 + 25.5iT - 2.19e3T^{2} \)
17 \( 1 - 88.9T + 4.91e3T^{2} \)
19 \( 1 - 54.2iT - 6.85e3T^{2} \)
23 \( 1 + 25.4iT - 1.21e4T^{2} \)
29 \( 1 - 231. iT - 2.43e4T^{2} \)
31 \( 1 + 117. iT - 2.97e4T^{2} \)
37 \( 1 - 392.T + 5.06e4T^{2} \)
41 \( 1 + 478.T + 6.89e4T^{2} \)
43 \( 1 + 253.T + 7.95e4T^{2} \)
47 \( 1 + 313.T + 1.03e5T^{2} \)
53 \( 1 - 558. iT - 1.48e5T^{2} \)
59 \( 1 + 258.T + 2.05e5T^{2} \)
61 \( 1 - 457. iT - 2.26e5T^{2} \)
67 \( 1 - 751.T + 3.00e5T^{2} \)
71 \( 1 - 297. iT - 3.57e5T^{2} \)
73 \( 1 - 302. iT - 3.89e5T^{2} \)
79 \( 1 + 488.T + 4.93e5T^{2} \)
83 \( 1 - 918.T + 5.71e5T^{2} \)
89 \( 1 - 61.4T + 7.04e5T^{2} \)
97 \( 1 - 175. iT - 9.12e5T^{2} \)
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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

−13.39395564466360686932785189307, −12.31868561793293920504284896737, −11.17212007217146194828478121897, −10.02955957663746875475454573169, −8.316671827610894062906456037978, −7.56828001104810212244846236320, −6.57289325924157799997328587500, −5.52042317030007940275865156099, −3.13998532347373394099481150081, −1.36610394421456075239381455616, 2.09705010972769571107263056157, 3.31964899932088951283215455750, 5.03074890315367748940999830104, 6.35908963265521375205670061990, 8.092126080113122651441795630454, 9.477703023394992796351988247920, 10.01683508640964461671031684159, 11.31396348214921514339226616601, 11.97095560806814314888455762251, 13.26445393110553327755960118218

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