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Properties

Label 2-105-105.59-c1-0-3
Degree $2$
Conductor $105$
Sign $0.988 - 0.153i$
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  + (0.322 + 0.558i)2-s + (−1.69 − 0.358i)3-s + (0.792 − 1.37i)4-s + (1.60 + 1.56i)5-s + (−0.346 − 1.06i)6-s + (2.64 + 0.105i)7-s + 2.31·8-s + (2.74 + 1.21i)9-s + (−0.354 + 1.39i)10-s + (−3.51 − 2.02i)11-s + (−1.83 + 2.04i)12-s − 4.21·13-s + (0.793 + 1.51i)14-s + (−2.15 − 3.21i)15-s + (−0.839 − 1.45i)16-s + (1.88 + 1.08i)17-s + ⋯
L(s)  = 1  + (0.227 + 0.394i)2-s + (−0.978 − 0.206i)3-s + (0.396 − 0.685i)4-s + (0.716 + 0.697i)5-s + (−0.141 − 0.433i)6-s + (0.999 + 0.0397i)7-s + 0.817·8-s + (0.914 + 0.404i)9-s + (−0.112 + 0.441i)10-s + (−1.05 − 0.611i)11-s + (−0.529 + 0.589i)12-s − 1.16·13-s + (0.212 + 0.403i)14-s + (−0.556 − 0.830i)15-s + (−0.209 − 0.363i)16-s + (0.457 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.988 - 0.153i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.988 - 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07555 + 0.0827929i\)
\(L(\frac12)\) \(\approx\) \(1.07555 + 0.0827929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.358i)T \)
5 \( 1 + (-1.60 - 1.56i)T \)
7 \( 1 + (-2.64 - 0.105i)T \)
good2 \( 1 + (-0.322 - 0.558i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3.51 + 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.21T + 13T^{2} \)
17 \( 1 + (-1.88 - 1.08i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.87 - 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.322 + 0.558i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 + 0.195i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.69 - 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 + 6.54iT - 43T^{2} \)
47 \( 1 + (6.75 - 3.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.60 - 6.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.66 + 9.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.05 + 3.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.56 - 4.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.13iT - 71T^{2} \)
73 \( 1 + (-2.61 + 4.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.87 + 3.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.27iT - 83T^{2} \)
89 \( 1 + (-0.447 - 0.774i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.89T + 97T^{2} \)
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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

−14.02160572997561743303882672026, −12.81150088257920980260342487524, −11.47916677248235808111968510286, −10.61112686206751738475219658481, −10.06359113296986825621859102373, −7.88669344856195060009567825630, −6.81844848781462582551565282773, −5.69897226240747832319658062036, −4.98144371842320715821640481892, −2.02337851965601909458889220254, 2.11768357566233185544331273952, 4.55540681064617519704862514331, 5.26220413539681490268758575800, 7.01506799851600345410450559935, 8.146704305495724404513368627869, 9.816572338572298069429840579782, 10.71954122160454164731168613384, 11.79041592939548977080935088532, 12.55350926577213789202778000531, 13.30112666289987877665261960788

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