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Properties

Label 2-105-35.4-c1-0-2
Degree $2$
Conductor $105$
Sign $0.550 - 0.834i$
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  + (−2.17 + 1.25i)2-s + (0.866 + 0.5i)3-s + (2.16 − 3.75i)4-s + (2.23 + 0.00136i)5-s − 2.51·6-s + (1.31 − 2.29i)7-s + 5.87i·8-s + (0.499 + 0.866i)9-s + (−4.87 + 2.81i)10-s + (−0.489 + 0.847i)11-s + (3.75 − 2.16i)12-s + 5.14i·13-s + (0.0275 + 6.65i)14-s + (1.93 + 1.11i)15-s + (−3.05 − 5.29i)16-s + (−3.59 − 2.07i)17-s + ⋯
L(s)  = 1  + (−1.54 + 0.889i)2-s + (0.499 + 0.288i)3-s + (1.08 − 1.87i)4-s + (0.999 + 0.000610i)5-s − 1.02·6-s + (0.496 − 0.868i)7-s + 2.07i·8-s + (0.166 + 0.288i)9-s + (−1.54 + 0.888i)10-s + (−0.147 + 0.255i)11-s + (1.08 − 0.625i)12-s + 1.42i·13-s + (0.00735 + 1.77i)14-s + (0.499 + 0.288i)15-s + (−0.763 − 1.32i)16-s + (−0.871 − 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.550 - 0.834i$
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: \((2,\ 105,\ (\ :1/2),\ 0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617005 + 0.332019i\)
\(L(\frac12)\) \(\approx\) \(0.617005 + 0.332019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.00136i)T \)
7 \( 1 + (-1.31 + 2.29i)T \)
good2 \( 1 + (2.17 - 1.25i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (0.489 - 0.847i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.14iT - 13T^{2} \)
17 \( 1 + (3.59 + 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.15 + 1.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.39 + 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 + (0.316 - 0.548i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.84 - 4.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 - 0.344iT - 43T^{2} \)
47 \( 1 + (-3.67 + 2.11i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.61 + 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.908 + 1.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.328 + 0.568i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.01 + 4.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 + (4.65 + 2.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.44 + 9.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.62iT - 83T^{2} \)
89 \( 1 + (-8.15 - 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.53iT - 97T^{2} \)
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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

−14.24428641304740243640484422482, −13.34881354931807437614096353835, −11.17411978355311648010853042106, −10.35496967939085209902065990059, −9.316190364556678984332818888882, −8.763253551360327463038417422700, −7.31330323550039300047253527218, −6.58388185222125151334966454148, −4.81565789455395266152381436401, −1.86548052668024560962857103361, 1.76346001999280288833826342344, 2.94360586972849811274325871876, 5.72964197772036066312807092387, 7.46976506605602729595379104281, 8.585108736588720789464984077492, 9.161604476395214285157026194463, 10.33307694386912846478849102363, 11.12195880488742527799887919123, 12.48078149419859808323692645977, 13.17786649948419645935230568597

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