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Properties

Label 2-105-5.2-c2-0-6
Degree $2$
Conductor $105$
Sign $0.423 - 0.906i$
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  + (2.41 + 2.41i)2-s + (1.22 − 1.22i)3-s + 7.68i·4-s + (4.18 − 2.74i)5-s + 5.92·6-s + (−1.87 − 1.87i)7-s + (−8.90 + 8.90i)8-s − 2.99i·9-s + (16.7 + 3.47i)10-s − 20.9·11-s + (9.40 + 9.40i)12-s + (−9.34 + 9.34i)13-s − 9.04i·14-s + (1.76 − 8.47i)15-s − 12.2·16-s + (7.08 + 7.08i)17-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (0.408 − 0.408i)3-s + 1.92i·4-s + (0.836 − 0.548i)5-s + 0.986·6-s + (−0.267 − 0.267i)7-s + (−1.11 + 1.11i)8-s − 0.333i·9-s + (1.67 + 0.347i)10-s − 1.90·11-s + (0.784 + 0.784i)12-s + (−0.718 + 0.718i)13-s − 0.645i·14-s + (0.117 − 0.565i)15-s − 0.768·16-s + (0.416 + 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.423 - 0.906i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.423 - 0.906i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27524 + 1.44869i\)
\(L(\frac12)\) \(\approx\) \(2.27524 + 1.44869i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-4.18 + 2.74i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-2.41 - 2.41i)T + 4iT^{2} \)
11 \( 1 + 20.9T + 121T^{2} \)
13 \( 1 + (9.34 - 9.34i)T - 169iT^{2} \)
17 \( 1 + (-7.08 - 7.08i)T + 289iT^{2} \)
19 \( 1 + 14.9iT - 361T^{2} \)
23 \( 1 + (-12.9 + 12.9i)T - 529iT^{2} \)
29 \( 1 - 39.6iT - 841T^{2} \)
31 \( 1 - 12.8T + 961T^{2} \)
37 \( 1 + (31.7 + 31.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 69.4T + 1.68e3T^{2} \)
43 \( 1 + (-4.46 + 4.46i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.41 + 4.41i)T + 2.20e3iT^{2} \)
53 \( 1 + (48.5 - 48.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 29.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.09T + 3.72e3T^{2} \)
67 \( 1 + (1.39 + 1.39i)T + 4.48e3iT^{2} \)
71 \( 1 + 15.9T + 5.04e3T^{2} \)
73 \( 1 + (-32.4 + 32.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (83.6 - 83.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 62.7iT - 7.92e3T^{2} \)
97 \( 1 + (85.4 + 85.4i)T + 9.40e3iT^{2} \)
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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

−13.80219820560027324775434381903, −12.86982204409400124173251614537, −12.55840631687303414624515075296, −10.49399737913839283705099958377, −9.024458795645925743736367181912, −7.78582583889129788743266355622, −6.85624248621104130967961645817, −5.59047950560194448326105865335, −4.66551532386580781972595069423, −2.74107191830797683522933022906, 2.40725196670167229761268255753, 3.15326896612943940512535457753, 4.99178253132589233973698800634, 5.77935658930368348780263784906, 7.79630764081569523186737594729, 9.815188435231095728308674272826, 10.18464599616167797957367840178, 11.20614153847878511595597189363, 12.57991444428888016348215566664, 13.25212365839498520003372010256

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