Properties

Label 2-105-105.83-c2-0-25
Degree $2$
Conductor $105$
Sign $-0.435 + 0.900i$
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  + (0.707 + 0.707i)2-s + (−0.854 − 2.87i)3-s − 3i·4-s + (−4.57 + 2.01i)5-s + (1.42 − 2.63i)6-s + (−5.77 − 3.94i)7-s + (4.94 − 4.94i)8-s + (−7.53 + 4.91i)9-s + (−4.65 − 1.81i)10-s − 2.58i·11-s + (−8.62 + 2.56i)12-s + (8.94 − 8.94i)13-s + (−1.29 − 6.87i)14-s + (9.69 + 11.4i)15-s − 4.99·16-s + (0.581 − 0.581i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.353i)2-s + (−0.284 − 0.958i)3-s − 0.750i·4-s + (−0.915 + 0.402i)5-s + (0.238 − 0.439i)6-s + (−0.825 − 0.564i)7-s + (0.618 − 0.618i)8-s + (−0.837 + 0.546i)9-s + (−0.465 − 0.181i)10-s − 0.235i·11-s + (−0.718 + 0.213i)12-s + (0.687 − 0.687i)13-s + (−0.0924 − 0.491i)14-s + (0.646 + 0.763i)15-s − 0.312·16-s + (0.0342 − 0.0342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.435 + 0.900i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.435 + 0.900i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.537658 - 0.857050i\)
\(L(\frac12)\) \(\approx\) \(0.537658 - 0.857050i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.854 + 2.87i)T \)
5 \( 1 + (4.57 - 2.01i)T \)
7 \( 1 + (5.77 + 3.94i)T \)
good2 \( 1 + (-0.707 - 0.707i)T + 4iT^{2} \)
11 \( 1 + 2.58iT - 121T^{2} \)
13 \( 1 + (-8.94 + 8.94i)T - 169iT^{2} \)
17 \( 1 + (-0.581 + 0.581i)T - 289iT^{2} \)
19 \( 1 - 16.5T + 361T^{2} \)
23 \( 1 + (-26.5 + 26.5i)T - 529iT^{2} \)
29 \( 1 + 11.5T + 841T^{2} \)
31 \( 1 - 30.8iT - 961T^{2} \)
37 \( 1 + (41.4 - 41.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 17.1T + 1.68e3T^{2} \)
43 \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-45.2 + 45.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-34.1 + 34.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 47.6iT - 3.48e3T^{2} \)
61 \( 1 + 78.9iT - 3.72e3T^{2} \)
67 \( 1 + (-72.4 + 72.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 49.0iT - 5.04e3T^{2} \)
73 \( 1 + (-26.0 + 26.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 75.8iT - 6.24e3T^{2} \)
83 \( 1 + (53.6 + 53.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 14.0iT - 7.92e3T^{2} \)
97 \( 1 + (25.9 + 25.9i)T + 9.40e3iT^{2} \)
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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

−13.33348179570851606855134764752, −12.31596076481823657240859206380, −11.05934597537128289841006317579, −10.31215025938571045663511953817, −8.529091656670237450245783684796, −7.14535410397616679544791953092, −6.58087011239751239142650295218, −5.21686618409417179869951748797, −3.35124504733449531519435395325, −0.72357733200806825756980922867, 3.22023649089247347666792604690, 4.10116427939317470885058391971, 5.44134118258553907755753535623, 7.22817697259111283402348790037, 8.674493517375120974848097251184, 9.452459602407238599129613192151, 11.09016717087511982605890558224, 11.72126453253888259989720832731, 12.58522358288910578437917910202, 13.65719600113482453404955081913

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