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Properties

Label 2-105-5.4-c3-0-12
Degree $2$
Conductor $105$
Sign $-0.907 + 0.420i$
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  − 3.33i·2-s − 3i·3-s − 3.14·4-s + (10.1 − 4.70i)5-s − 10.0·6-s − 7i·7-s − 16.2i·8-s − 9·9-s + (−15.7 − 33.8i)10-s + 18.3·11-s + 9.42i·12-s + 10.1i·13-s − 23.3·14-s + (−14.1 − 30.4i)15-s − 79.2·16-s + 24.6i·17-s + ⋯
L(s)  = 1  − 1.18i·2-s − 0.577i·3-s − 0.392·4-s + (0.907 − 0.420i)5-s − 0.681·6-s − 0.377i·7-s − 0.716i·8-s − 0.333·9-s + (−0.496 − 1.07i)10-s + 0.502·11-s + 0.226i·12-s + 0.216i·13-s − 0.446·14-s + (−0.242 − 0.523i)15-s − 1.23·16-s + 0.352i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.397949 - 1.80396i\)
\(L(\frac12)\) \(\approx\) \(0.397949 - 1.80396i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (-10.1 + 4.70i)T \)
7 \( 1 + 7iT \)
good2 \( 1 + 3.33iT - 8T^{2} \)
11 \( 1 - 18.3T + 1.33e3T^{2} \)
13 \( 1 - 10.1iT - 2.19e3T^{2} \)
17 \( 1 - 24.6iT - 4.91e3T^{2} \)
19 \( 1 + 77.4T + 6.85e3T^{2} \)
23 \( 1 - 149. iT - 1.21e4T^{2} \)
29 \( 1 - 10.2T + 2.43e4T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + 215. iT - 5.06e4T^{2} \)
41 \( 1 - 495.T + 6.89e4T^{2} \)
43 \( 1 + 220. iT - 7.95e4T^{2} \)
47 \( 1 + 212. iT - 1.03e5T^{2} \)
53 \( 1 - 532. iT - 1.48e5T^{2} \)
59 \( 1 - 324.T + 2.05e5T^{2} \)
61 \( 1 - 653.T + 2.26e5T^{2} \)
67 \( 1 - 819. iT - 3.00e5T^{2} \)
71 \( 1 + 466.T + 3.57e5T^{2} \)
73 \( 1 - 173. iT - 3.89e5T^{2} \)
79 \( 1 + 810.T + 4.93e5T^{2} \)
83 \( 1 - 12.3iT - 5.71e5T^{2} \)
89 \( 1 - 33.8T + 7.04e5T^{2} \)
97 \( 1 - 810. iT - 9.12e5T^{2} \)
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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

−12.78549561616004261542332361739, −11.83732156678396125558326077722, −10.79982667499988843290559703660, −9.801468876074160093649573506054, −8.797657922814876421840922403011, −7.11550841786074924207659153327, −5.91609950644653285535866691362, −4.06723043474538779451258251349, −2.31979613775234766593493356426, −1.12900717035133115853167033658, 2.54522885983110857556360337603, 4.71700071546656835211690477987, 5.98208022610200834238466625826, 6.69455989614963843654911719099, 8.234710777355581476932661941799, 9.236386965188742344004833572079, 10.40116614385215130025285165578, 11.47597931370435124715420138405, 12.97806816376632636969181926835, 14.32962776154690745749925068643

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