Properties

Label 2-105-7.4-c1-0-5
Degree $2$
Conductor $105$
Sign $0.126 + 0.991i$
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  + (−0.366 − 0.633i)2-s + (0.5 − 0.866i)3-s + (0.732 − 1.26i)4-s + (−0.5 − 0.866i)5-s − 0.732·6-s + (−0.866 + 2.5i)7-s − 2.53·8-s + (−0.499 − 0.866i)9-s + (−0.366 + 0.633i)10-s + (1.36 − 2.36i)11-s + (−0.732 − 1.26i)12-s + 5.73·13-s + (1.90 − 0.366i)14-s − 0.999·15-s + (−0.535 − 0.928i)16-s + (−3.36 + 5.83i)17-s + ⋯
L(s)  = 1  + (−0.258 − 0.448i)2-s + (0.288 − 0.499i)3-s + (0.366 − 0.633i)4-s + (−0.223 − 0.387i)5-s − 0.298·6-s + (−0.327 + 0.944i)7-s − 0.896·8-s + (−0.166 − 0.288i)9-s + (−0.115 + 0.200i)10-s + (0.411 − 0.713i)11-s + (−0.211 − 0.366i)12-s + 1.58·13-s + (0.508 − 0.0978i)14-s − 0.258·15-s + (−0.133 − 0.232i)16-s + (−0.816 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.126 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752416 - 0.662617i\)
\(L(\frac12)\) \(\approx\) \(0.752416 - 0.662617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good2 \( 1 + (0.366 + 0.633i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.73T + 13T^{2} \)
17 \( 1 + (3.36 - 5.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 2.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 + 7.19T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.19 + 7.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.09 - 8.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.33 - 2.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 + (-2.33 + 4.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.69 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + (-4.56 - 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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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

−13.39749308982028142947665312816, −12.36451433380787406288041599613, −11.45649471800004331812478790301, −10.48238865047600500967617778295, −8.928652244357981869079406491368, −8.530060341764745914046822776734, −6.50858148979081882075054991466, −5.75405956470893578673517064721, −3.42465522315660724767131844910, −1.60408273106223122148677741851, 3.09315354450759599404135906495, 4.32094839033757970734898059383, 6.49993823231428821026950099325, 7.28193823974852025302801968297, 8.513907458480985139169851277761, 9.577915929829843295969819615190, 10.89098960203284352188763419662, 11.70675238767127503258581400763, 13.18491854728157339603073539450, 14.02023374441024965819421090205

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