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Properties

Label 2-105-35.2-c2-0-6
Degree $2$
Conductor $105$
Sign $0.598 - 0.801i$
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  + (−2.87 + 0.770i)2-s + (1.67 + 0.448i)3-s + (4.20 − 2.42i)4-s + (3.78 + 3.26i)5-s − 5.15·6-s + (−1.65 − 6.80i)7-s + (−1.79 + 1.79i)8-s + (2.59 + 1.50i)9-s + (−13.3 − 6.46i)10-s + (6.96 + 12.0i)11-s + (8.11 − 2.17i)12-s + (7.79 − 7.79i)13-s + (9.99 + 18.2i)14-s + (4.87 + 7.15i)15-s + (−5.93 + 10.2i)16-s + (−6.92 + 25.8i)17-s + ⋯
L(s)  = 1  + (−1.43 + 0.385i)2-s + (0.557 + 0.149i)3-s + (1.05 − 0.606i)4-s + (0.757 + 0.652i)5-s − 0.858·6-s + (−0.236 − 0.971i)7-s + (−0.223 + 0.223i)8-s + (0.288 + 0.166i)9-s + (−1.33 − 0.646i)10-s + (0.632 + 1.09i)11-s + (0.676 − 0.181i)12-s + (0.599 − 0.599i)13-s + (0.713 + 1.30i)14-s + (0.324 + 0.477i)15-s + (−0.370 + 0.642i)16-s + (−0.407 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.598 - 0.801i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.824649 + 0.413476i\)
\(L(\frac12)\) \(\approx\) \(0.824649 + 0.413476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (-3.78 - 3.26i)T \)
7 \( 1 + (1.65 + 6.80i)T \)
good2 \( 1 + (2.87 - 0.770i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-6.96 - 12.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.79 + 7.79i)T - 169iT^{2} \)
17 \( 1 + (6.92 - 25.8i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-28.5 - 16.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.48 + 20.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 18.1iT - 841T^{2} \)
31 \( 1 + (11.5 + 20.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (20.1 - 5.40i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 1.69T + 1.68e3T^{2} \)
43 \( 1 + (-26.7 + 26.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (10.2 - 2.75i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (43.0 + 11.5i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-9.04 + 5.22i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-40.5 + 70.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-10.9 + 40.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 20.1T + 5.04e3T^{2} \)
73 \( 1 + (-6.28 - 1.68i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-10.5 - 6.08i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-20.6 + 20.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (145. + 83.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (66.3 + 66.3i)T + 9.40e3iT^{2} \)
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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

−13.87737575184136521003469281365, −12.80158834420777331093196665424, −10.88305128707956000279413816215, −10.09066492998362562055378004672, −9.552255681687758658859002188265, −8.212326791596604359902798775176, −7.25365048889526322476926819350, −6.25652670870400691469965841393, −3.84134473613635722717382155746, −1.65285406139374103032780829783, 1.26531821781736930854859382561, 2.84625686489290367040663566026, 5.36259193997882318844284873748, 6.94849895849385170636340595124, 8.452344160852883839859410548551, 9.247989527283403932846593469822, 9.463324215182460979279331777165, 11.19600390834070602913256866655, 11.96442626963193735465032378444, 13.51848044324976539006033402784

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