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Properties

Label 2-105-105.32-c1-0-6
Degree $2$
Conductor $105$
Sign $0.770 - 0.637i$
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  + (2.35 + 0.631i)2-s + (−1.54 + 0.775i)3-s + (3.42 + 1.97i)4-s + (−0.0540 − 2.23i)5-s + (−4.13 + 0.849i)6-s + (−1.91 + 1.82i)7-s + (3.36 + 3.36i)8-s + (1.79 − 2.40i)9-s + (1.28 − 5.30i)10-s + (−3.08 − 1.77i)11-s + (−6.83 − 0.406i)12-s + (1.28 − 1.28i)13-s + (−5.67 + 3.08i)14-s + (1.81 + 3.42i)15-s + (1.85 + 3.21i)16-s + (0.792 + 2.95i)17-s + ⋯
L(s)  = 1  + (1.66 + 0.446i)2-s + (−0.894 + 0.447i)3-s + (1.71 + 0.987i)4-s + (−0.0241 − 0.999i)5-s + (−1.68 + 0.346i)6-s + (−0.725 + 0.688i)7-s + (1.19 + 1.19i)8-s + (0.599 − 0.800i)9-s + (0.406 − 1.67i)10-s + (−0.928 − 0.536i)11-s + (−1.97 − 0.117i)12-s + (0.356 − 0.356i)13-s + (−1.51 + 0.823i)14-s + (0.469 + 0.883i)15-s + (0.463 + 0.803i)16-s + (0.192 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62806 + 0.585717i\)
\(L(\frac12)\) \(\approx\) \(1.62806 + 0.585717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 0.775i)T \)
5 \( 1 + (0.0540 + 2.23i)T \)
7 \( 1 + (1.91 - 1.82i)T \)
good2 \( 1 + (-2.35 - 0.631i)T + (1.73 + i)T^{2} \)
11 \( 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \)
17 \( 1 + (-0.792 - 2.95i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.331 + 0.191i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.658 - 2.45i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.34 - 5.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 - 0.335i)T - 43iT^{2} \)
47 \( 1 + (-2.80 - 0.751i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.04 - 0.815i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.81 + 6.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.3 + 3.31i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (0.849 + 3.17i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.973 + 0.973i)T + 83iT^{2} \)
89 \( 1 + (-1.51 - 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 + 10.3i)T + 97iT^{2} \)
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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

−13.60261886039178459235992854933, −12.80601086764414156635374259021, −12.25166257730708830343215997665, −11.20935306810921370871956843532, −9.780680426967857996457038678964, −8.180577503758516316971543172110, −6.38156973014447193831862892205, −5.63148947963667615192455781611, −4.77446497453703299763703119885, −3.35825673050670158052135747612, 2.57840070936464811318725115299, 4.11073559390851045737695272659, 5.47937492612628641895201661495, 6.60236395183664917839902528809, 7.31294601876405843772222570020, 10.21634446088090384657214377407, 10.76168491432123600880255555463, 11.82572555280642318453772648011, 12.66388240440585691984774421719, 13.58056449639412908991287292703

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