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Properties

Label 2-105-21.20-c1-0-3
Degree $2$
Conductor $105$
Sign $-0.586 - 0.810i$
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  + 2.52i·2-s + (1.68 + 0.396i)3-s − 4.37·4-s + 5-s + (−1 + 4.25i)6-s + (−2 − 1.73i)7-s − 5.98i·8-s + (2.68 + 1.33i)9-s + 2.52i·10-s + 0.792i·11-s + (−7.37 − 1.73i)12-s − 5.84i·13-s + (4.37 − 5.04i)14-s + (1.68 + 0.396i)15-s + 6.37·16-s − 1.37·17-s + ⋯
L(s)  = 1  + 1.78i·2-s + (0.973 + 0.228i)3-s − 2.18·4-s + 0.447·5-s + (−0.408 + 1.73i)6-s + (−0.755 − 0.654i)7-s − 2.11i·8-s + (0.895 + 0.445i)9-s + 0.798i·10-s + 0.238i·11-s + (−2.12 − 0.500i)12-s − 1.61i·13-s + (1.16 − 1.34i)14-s + (0.435 + 0.102i)15-s + 1.59·16-s − 0.332·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.586 - 0.810i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548057 + 1.07296i\)
\(L(\frac12)\) \(\approx\) \(0.548057 + 1.07296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.396i)T \)
5 \( 1 - T \)
7 \( 1 + (2 + 1.73i)T \)
good2 \( 1 - 2.52iT - 2T^{2} \)
11 \( 1 - 0.792iT - 11T^{2} \)
13 \( 1 + 5.84iT - 13T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 - 4.25iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6.74T + 43T^{2} \)
47 \( 1 + 7.37T + 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 + 6.74T + 67T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 - 5.48T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 - 1.08iT - 97T^{2} \)
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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

−14.43456139951680759318943733384, −13.32710682136718821553718083859, −12.97623666769740754650107590850, −10.27858515557067864073500418053, −9.574581561373486710942169590745, −8.354251635499284084465239115798, −7.53853794170120613678327812209, −6.43395514024750228163581171401, −5.08898604836189796432940207717, −3.53528712972967195973977795790, 1.97639919723049730284999905793, 3.07061117435097738365517582810, 4.47114510218065475631758746190, 6.61999853750819441477048005630, 8.675881314756184674941142998726, 9.274234105409606876667414574047, 10.07184794099349296693698259091, 11.45651902139635798312840509276, 12.34331422317139262127515153827, 13.34326477466239129697430592494

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