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Properties

Label 2-105-105.62-c2-0-24
Degree $2$
Conductor $105$
Sign $0.0732 + 0.997i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.28 − 2.28i)2-s + (2.80 − 1.07i)3-s − 6.40i·4-s + (−1.80 + 4.66i)5-s + (3.93 − 8.83i)6-s + (−6.22 − 3.20i)7-s + (−5.48 − 5.48i)8-s + (6.69 − 6.01i)9-s + (6.50 + 14.7i)10-s + 11.1i·11-s + (−6.88 − 17.9i)12-s + (5.82 + 5.82i)13-s + (−21.5 + 6.86i)14-s + (−0.0592 + 14.9i)15-s + 0.592·16-s + (−6.84 − 6.84i)17-s + ⋯
L(s)  = 1  + (1.14 − 1.14i)2-s + (0.933 − 0.358i)3-s − 1.60i·4-s + (−0.361 + 0.932i)5-s + (0.656 − 1.47i)6-s + (−0.888 − 0.458i)7-s + (−0.685 − 0.685i)8-s + (0.743 − 0.668i)9-s + (0.650 + 1.47i)10-s + 1.01i·11-s + (−0.573 − 1.49i)12-s + (0.448 + 0.448i)13-s + (−1.53 + 0.490i)14-s + (−0.00395 + 0.999i)15-s + 0.0370·16-s + (−0.402 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0732 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0732 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.0732 + 0.997i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.0732 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.97462 - 1.83482i\)
\(L(\frac12)\) \(\approx\) \(1.97462 - 1.83482i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.80 + 1.07i)T \)
5 \( 1 + (1.80 - 4.66i)T \)
7 \( 1 + (6.22 + 3.20i)T \)
good2 \( 1 + (-2.28 + 2.28i)T - 4iT^{2} \)
11 \( 1 - 11.1iT - 121T^{2} \)
13 \( 1 + (-5.82 - 5.82i)T + 169iT^{2} \)
17 \( 1 + (6.84 + 6.84i)T + 289iT^{2} \)
19 \( 1 + 25.0T + 361T^{2} \)
23 \( 1 + (-23.3 - 23.3i)T + 529iT^{2} \)
29 \( 1 + 10.6T + 841T^{2} \)
31 \( 1 + 26.9iT - 961T^{2} \)
37 \( 1 + (20.8 + 20.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 32.9T + 1.68e3T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 1.84e3iT^{2} \)
47 \( 1 + (-59.1 - 59.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (26.0 + 26.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 70.2iT - 3.48e3T^{2} \)
61 \( 1 + 14.1iT - 3.72e3T^{2} \)
67 \( 1 + (6.14 + 6.14i)T + 4.48e3iT^{2} \)
71 \( 1 - 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (51.1 + 51.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 16.8iT - 6.24e3T^{2} \)
83 \( 1 + (-31.3 + 31.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 70.8iT - 7.92e3T^{2} \)
97 \( 1 + (114. - 114. i)T - 9.40e3iT^{2} \)
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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

−13.21346624565212350044401623726, −12.51885300100396644318178720296, −11.32638146085163797966125429150, −10.33597945157213224408215914992, −9.313052542443474363119361210001, −7.46038932405881952406123350021, −6.46525009093791215678625068838, −4.27050028884245544145079001928, −3.37266991360289495145448721551, −2.15253824130510070704307976764, 3.24655359278124167274195254157, 4.34651583178621076859912448243, 5.62549018494909449251991232971, 6.86604488332294751976078218433, 8.433526730842855256247195947092, 8.769038683700180704615070322394, 10.53545406780469494312480115538, 12.40683251055817468642602598645, 13.09559127279759363851469572740, 13.71524001708872038269496044527

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