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Properties

Label 2-105-35.4-c1-0-5
Degree $2$
Conductor $105$
Sign $0.631 + 0.775i$
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  + (1.34 − 0.776i)2-s + (−0.866 − 0.5i)3-s + (0.204 − 0.355i)4-s + (1.88 − 1.19i)5-s − 1.55·6-s + (−0.478 − 2.60i)7-s + 2.46i·8-s + (0.499 + 0.866i)9-s + (1.61 − 3.07i)10-s + (−2.21 + 3.83i)11-s + (−0.355 + 0.204i)12-s + 1.73i·13-s + (−2.66 − 3.12i)14-s + (−2.23 + 0.0917i)15-s + (2.32 + 4.02i)16-s + (−2.36 − 1.36i)17-s + ⋯
L(s)  = 1  + (0.950 − 0.548i)2-s + (−0.499 − 0.288i)3-s + (0.102 − 0.177i)4-s + (0.844 − 0.535i)5-s − 0.633·6-s + (−0.180 − 0.983i)7-s + 0.872i·8-s + (0.166 + 0.288i)9-s + (0.509 − 0.972i)10-s + (−0.667 + 1.15i)11-s + (−0.102 + 0.0591i)12-s + 0.480i·13-s + (−0.711 − 0.835i)14-s + (−0.576 + 0.0236i)15-s + (0.581 + 1.00i)16-s + (−0.573 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28000 - 0.608641i\)
\(L(\frac12)\) \(\approx\) \(1.28000 - 0.608641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.88 + 1.19i)T \)
7 \( 1 + (0.478 + 2.60i)T \)
good2 \( 1 + (-1.34 + 0.776i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.21 - 3.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (2.36 + 1.36i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.152 + 0.264i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.08 - 3.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.79T + 29T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.18 + 1.83i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 + 9.71iT - 43T^{2} \)
47 \( 1 + (1.57 - 0.908i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.48 - 0.857i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.571 - 0.989i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.26 + 4.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-11.1 - 6.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.35 - 5.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.09iT - 83T^{2} \)
89 \( 1 + (-2.03 - 3.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.87iT - 97T^{2} \)
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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

−13.59079004215671317901868896040, −12.65556594805957045076094294084, −11.89127377565386976127065803425, −10.61133134523084751239454646881, −9.669758107271854260171177213114, −7.987015455236908921999439068147, −6.57949386212705154386916514929, −5.15481733353187785008500863390, −4.24566316324407391329028298964, −2.13593747941654210264596408384, 3.01438337874772800765663162593, 4.89535404242230675659989411303, 5.95956921456551317837164483426, 6.42297906415929085457156988520, 8.429246120178855468715711404730, 9.864495511542898615517966328942, 10.68206833691557536474888581137, 12.09140080558479808507886285950, 13.13066646825401737962636695500, 13.93390984843127097183693833274

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