Properties

Label 2-105-105.23-c1-0-2
Degree $2$
Conductor $105$
Sign $0.860 - 0.509i$
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  + (−1.46 + 0.391i)2-s + (1.49 − 0.879i)3-s + (0.246 − 0.142i)4-s + (−0.207 + 2.22i)5-s + (−1.83 + 1.86i)6-s + (2.36 + 1.17i)7-s + (1.83 − 1.83i)8-s + (1.45 − 2.62i)9-s + (−0.567 − 3.33i)10-s + (−0.791 + 0.457i)11-s + (0.243 − 0.429i)12-s + (3.07 + 3.07i)13-s + (−3.92 − 0.791i)14-s + (1.64 + 3.50i)15-s + (−2.24 + 3.88i)16-s + (0.311 − 1.16i)17-s + ⋯
L(s)  = 1  + (−1.03 + 0.276i)2-s + (0.861 − 0.507i)3-s + (0.123 − 0.0712i)4-s + (−0.0929 + 0.995i)5-s + (−0.749 + 0.762i)6-s + (0.895 + 0.444i)7-s + (0.648 − 0.648i)8-s + (0.484 − 0.874i)9-s + (−0.179 − 1.05i)10-s + (−0.238 + 0.137i)11-s + (0.0702 − 0.124i)12-s + (0.854 + 0.854i)13-s + (−1.04 − 0.211i)14-s + (0.425 + 0.905i)15-s + (−0.561 + 0.971i)16-s + (0.0755 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778710 + 0.213469i\)
\(L(\frac12)\) \(\approx\) \(0.778710 + 0.213469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.49 + 0.879i)T \)
5 \( 1 + (0.207 - 2.22i)T \)
7 \( 1 + (-2.36 - 1.17i)T \)
good2 \( 1 + (1.46 - 0.391i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \)
17 \( 1 + (-0.311 + 1.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.95 + 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.505 + 1.88i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.207 - 0.774i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (-10.1 + 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (10.6 + 2.85i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (0.564 - 2.10i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.62 + 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.38 + 2.38i)T - 83iT^{2} \)
89 \( 1 + (-5.64 + 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−13.97868878897746856923940573401, −13.03111896231221299551573921741, −11.54224342120299500546981494748, −10.49464886770197310644675178294, −9.176179365354942359036782287098, −8.437453028826817647129067522572, −7.46806916712597404758965733486, −6.51215534782253831858188002883, −4.07426869154700117392793165439, −2.12400642151089781795677030840, 1.62700938985477079762534734678, 4.04405839665633709146452292090, 5.27962775951509313949418947331, 7.925622455267189200437780252487, 8.271531537973568583866600883295, 9.213576198273443383511374104965, 10.37744975470307876453951565138, 11.04249291441310211086197178559, 12.77209802507933055247881066000, 13.72193250691506391552183513742

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