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Properties

Label 2-105-35.17-c1-0-4
Degree $2$
Conductor $105$
Sign $0.456 - 0.889i$
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  + (−0.401 + 1.49i)2-s + (0.965 − 0.258i)3-s + (−0.346 − 0.200i)4-s + (2.14 − 0.625i)5-s + 1.54i·6-s + (−1.01 − 2.44i)7-s + (−1.75 + 1.75i)8-s + (0.866 − 0.499i)9-s + (0.0747 + 3.46i)10-s + (−2.59 + 4.49i)11-s + (−0.386 − 0.103i)12-s + (−3.30 − 3.30i)13-s + (4.06 − 0.545i)14-s + (1.91 − 1.15i)15-s + (−2.32 − 4.01i)16-s + (0.00519 + 0.0194i)17-s + ⋯
L(s)  = 1  + (−0.283 + 1.05i)2-s + (0.557 − 0.149i)3-s + (−0.173 − 0.100i)4-s + (0.960 − 0.279i)5-s + 0.632i·6-s + (−0.385 − 0.922i)7-s + (−0.619 + 0.619i)8-s + (0.288 − 0.166i)9-s + (0.0236 + 1.09i)10-s + (−0.782 + 1.35i)11-s + (−0.111 − 0.0299i)12-s + (−0.917 − 0.917i)13-s + (1.08 − 0.145i)14-s + (0.493 − 0.299i)15-s + (−0.580 − 1.00i)16-s + (0.00126 + 0.00470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968553 + 0.591970i\)
\(L(\frac12)\) \(\approx\) \(0.968553 + 0.591970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-2.14 + 0.625i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
good2 \( 1 + (0.401 - 1.49i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (2.59 - 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.30 + 3.30i)T + 13iT^{2} \)
17 \( 1 + (-0.00519 - 0.0194i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.24 - 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.24 + 0.601i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 - 3.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.714 - 2.66i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.68iT - 41T^{2} \)
43 \( 1 + (2.79 - 2.79i)T - 43iT^{2} \)
47 \( 1 + (-1.13 - 0.303i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.23 - 4.60i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.222 - 0.385i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 0.684i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.70 + 1.52i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.14T + 71T^{2} \)
73 \( 1 + (-7.15 + 1.91i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.47 - 2.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.77 + 3.77i)T + 83iT^{2} \)
89 \( 1 + (-1.91 - 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 10.5i)T - 97iT^{2} \)
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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

−14.09875057172075236566491304550, −13.15859203457444051758100348110, −12.21410413303572847719452124049, −10.15408377391395638055212508795, −9.722573276080398303447218451718, −8.113295787301535064311025171352, −7.40962895824589977391971225629, −6.27487273487002282369475690312, −4.86234762751039754877490148627, −2.55560924770498651219063149043, 2.22405775237366614160234228595, 3.14138796617204545479242838466, 5.46195069634873179661153309379, 6.75033810950029897045487110362, 8.687131131374855565778281146531, 9.453610640509007899174329925489, 10.32332565060281735225312115354, 11.35639819576601145548043948772, 12.46352948526734061076659577893, 13.46315448759606462082676451706

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