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Properties

Label 2-105-105.104-c3-0-10
Degree $2$
Conductor $105$
Sign $-0.588 - 0.808i$
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  − 4.45·2-s + (5.00 + 1.40i)3-s + 11.8·4-s + (0.892 + 11.1i)5-s + (−22.3 − 6.26i)6-s + (−12.5 + 13.5i)7-s − 17.3·8-s + (23.0 + 14.0i)9-s + (−3.98 − 49.7i)10-s − 30.2i·11-s + (59.4 + 16.6i)12-s − 18.9·13-s + (56.1 − 60.5i)14-s + (−11.1 + 57.0i)15-s − 17.7·16-s + 4.59i·17-s + ⋯
L(s)  = 1  − 1.57·2-s + (0.962 + 0.270i)3-s + 1.48·4-s + (0.0798 + 0.996i)5-s + (−1.51 − 0.426i)6-s + (−0.680 + 0.733i)7-s − 0.766·8-s + (0.853 + 0.520i)9-s + (−0.125 − 1.57i)10-s − 0.830i·11-s + (1.43 + 0.401i)12-s − 0.404·13-s + (1.07 − 1.15i)14-s + (−0.192 + 0.981i)15-s − 0.277·16-s + 0.0655i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.338465 + 0.665057i\)
\(L(\frac12)\) \(\approx\) \(0.338465 + 0.665057i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.00 - 1.40i)T \)
5 \( 1 + (-0.892 - 11.1i)T \)
7 \( 1 + (12.5 - 13.5i)T \)
good2 \( 1 + 4.45T + 8T^{2} \)
11 \( 1 + 30.2iT - 1.33e3T^{2} \)
13 \( 1 + 18.9T + 2.19e3T^{2} \)
17 \( 1 - 4.59iT - 4.91e3T^{2} \)
19 \( 1 - 119. iT - 6.85e3T^{2} \)
23 \( 1 + 134.T + 1.21e4T^{2} \)
29 \( 1 - 203. iT - 2.43e4T^{2} \)
31 \( 1 - 61.4iT - 2.97e4T^{2} \)
37 \( 1 + 337. iT - 5.06e4T^{2} \)
41 \( 1 - 135.T + 6.89e4T^{2} \)
43 \( 1 - 270. iT - 7.95e4T^{2} \)
47 \( 1 - 273. iT - 1.03e5T^{2} \)
53 \( 1 + 222.T + 1.48e5T^{2} \)
59 \( 1 - 735.T + 2.05e5T^{2} \)
61 \( 1 - 312. iT - 2.26e5T^{2} \)
67 \( 1 + 751. iT - 3.00e5T^{2} \)
71 \( 1 + 640. iT - 3.57e5T^{2} \)
73 \( 1 - 469.T + 3.89e5T^{2} \)
79 \( 1 - 126.T + 4.93e5T^{2} \)
83 \( 1 - 299. iT - 5.71e5T^{2} \)
89 \( 1 - 425.T + 7.04e5T^{2} \)
97 \( 1 - 561.T + 9.12e5T^{2} \)
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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

−13.96839457254539312283320196046, −12.44235280612870713023052824772, −10.97382775634634586660493230111, −10.10471808969334140196648827373, −9.372452652846232962998495579412, −8.348814126865286599573559068108, −7.44384341936582632610466171445, −6.14041615174872392524998271261, −3.41602223137723072383888040449, −2.12019583639682809479427387381, 0.60050790810072583704116498473, 2.18947290520628856205670312843, 4.33054322811558650706039872836, 6.76991543470199521690499741443, 7.68822273911436371628595509351, 8.619508960890857983975519838734, 9.674326796435430706922396771766, 10.00149258150682641976065307195, 11.77143377239723704337970444846, 13.00026177137765625207342955686

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