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Properties

Label 2-105-35.2-c2-0-8
Degree $2$
Conductor $105$
Sign $0.918 - 0.395i$
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  + (−0.435 + 0.116i)2-s + (1.67 + 0.448i)3-s + (−3.28 + 1.89i)4-s + (3.62 − 3.44i)5-s − 0.780·6-s + (6.87 + 1.29i)7-s + (2.48 − 2.48i)8-s + (2.59 + 1.50i)9-s + (−1.17 + 1.92i)10-s + (10.0 + 17.3i)11-s + (−6.35 + 1.70i)12-s + (−2.67 + 2.67i)13-s + (−3.14 + 0.240i)14-s + (7.60 − 4.13i)15-s + (6.80 − 11.7i)16-s + (3.22 − 12.0i)17-s + ⋯
L(s)  = 1  + (−0.217 + 0.0583i)2-s + (0.557 + 0.149i)3-s + (−0.822 + 0.474i)4-s + (0.724 − 0.689i)5-s − 0.130·6-s + (0.982 + 0.184i)7-s + (0.310 − 0.310i)8-s + (0.288 + 0.166i)9-s + (−0.117 + 0.192i)10-s + (0.910 + 1.57i)11-s + (−0.529 + 0.141i)12-s + (−0.205 + 0.205i)13-s + (−0.224 + 0.0171i)14-s + (0.507 − 0.275i)15-s + (0.425 − 0.736i)16-s + (0.189 − 0.708i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.918 - 0.395i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44779 + 0.298174i\)
\(L(\frac12)\) \(\approx\) \(1.44779 + 0.298174i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (-3.62 + 3.44i)T \)
7 \( 1 + (-6.87 - 1.29i)T \)
good2 \( 1 + (0.435 - 0.116i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-10.0 - 17.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.67 - 2.67i)T - 169iT^{2} \)
17 \( 1 + (-3.22 + 12.0i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (18.7 + 10.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.07 - 26.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 29.0iT - 841T^{2} \)
31 \( 1 + (22.1 + 38.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (59.0 - 15.8i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 19.8T + 1.68e3T^{2} \)
43 \( 1 + (14.0 - 14.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (10.8 - 2.90i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (31.2 + 8.37i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-49.6 + 28.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.82 - 17.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.54 - 13.2i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 71.1T + 5.04e3T^{2} \)
73 \( 1 + (-90.0 - 24.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (112. + 64.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (20.5 - 20.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (29.0 + 16.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (36.1 + 36.1i)T + 9.40e3iT^{2} \)
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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

−13.62533687272666342579726416391, −12.70542695893549464521605148644, −11.69364686810469921205091613935, −9.837699917441167727928824273870, −9.298222462976073248895003210839, −8.340089323213183657874032933192, −7.17094521156235568560258126578, −5.07281909049204642787958016919, −4.23562683515374211207661577763, −1.87216887616669924065860982998, 1.57360604176926604703896202255, 3.65863725277374215074581682007, 5.35612413068966955975762495789, 6.63612023387007300731362571235, 8.382864571566544298970934885681, 8.883356774245701191269123189162, 10.38541434223993328468628028297, 10.91191272681252695702170407316, 12.63129960695401061305671154674, 13.90038478629025765187971721904

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