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Properties

Label 2-105-35.4-c1-0-6
Degree $2$
Conductor $105$
Sign $0.844 + 0.536i$
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  + (1.54 − 0.890i)2-s + (0.866 + 0.5i)3-s + (0.587 − 1.01i)4-s + (−1.33 − 1.79i)5-s + 1.78·6-s + (−2.40 + 1.09i)7-s + 1.47i·8-s + (0.499 + 0.866i)9-s + (−3.66 − 1.56i)10-s + (1.03 − 1.80i)11-s + (1.01 − 0.587i)12-s + 3.13i·13-s + (−2.74 + 3.83i)14-s + (−0.264 − 2.22i)15-s + (2.48 + 4.30i)16-s + (−1.84 − 1.06i)17-s + ⋯
L(s)  = 1  + (1.09 − 0.629i)2-s + (0.499 + 0.288i)3-s + (0.293 − 0.508i)4-s + (−0.598 − 0.800i)5-s + 0.727·6-s + (−0.910 + 0.413i)7-s + 0.520i·8-s + (0.166 + 0.288i)9-s + (−1.15 − 0.496i)10-s + (0.313 − 0.542i)11-s + (0.293 − 0.169i)12-s + 0.868i·13-s + (−0.732 + 1.02i)14-s + (−0.0682 − 0.573i)15-s + (0.621 + 1.07i)16-s + (−0.447 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.844 + 0.536i$
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: \((2,\ 105,\ (\ :1/2),\ 0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58011 - 0.459295i\)
\(L(\frac12)\) \(\approx\) \(1.58011 - 0.459295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.79i)T \)
7 \( 1 + (2.40 - 1.09i)T \)
good2 \( 1 + (-1.54 + 0.890i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.03 + 1.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
17 \( 1 + (1.84 + 1.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.86 + 6.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.79 + 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.01T + 29T^{2} \)
31 \( 1 + (1.45 - 2.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.04 - 1.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.99T + 41T^{2} \)
43 \( 1 + 4.99iT - 43T^{2} \)
47 \( 1 + (2.11 - 1.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.58 + 4.95i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.47 + 2.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.44 - 9.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.32 - 1.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + (-7.40 - 4.27i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.05 - 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.75iT - 83T^{2} \)
89 \( 1 + (-0.309 - 0.535i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.296iT - 97T^{2} \)
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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

−13.38197009788108638785737944016, −12.82000086575638079220878361158, −11.81712403356698454969410398221, −10.90548334873308134976732814095, −9.137809677414155165436808923067, −8.607233655424823182251787299074, −6.69849696537146143897274029132, −5.00123589544040823823887086210, −4.02584850869289944909730788564, −2.73014667258975010211173949989, 3.20362675685399739141785184270, 4.18813261495358227722058174304, 6.06989773036214069932555838220, 6.92987884473997185158720026399, 7.901488531876882165170408973083, 9.650816917211696717326942789161, 10.69590500994696857000186951707, 12.37701415398667568778917417791, 12.93486915864833196306195606065, 14.02214352079173090884014212022

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