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Properties

Label 2-105-35.12-c1-0-1
Degree $2$
Conductor $105$
Sign $0.548 - 0.836i$
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  + (−2.42 + 0.650i)2-s + (−0.258 + 0.965i)3-s + (3.73 − 2.15i)4-s + (0.780 − 2.09i)5-s − 2.51i·6-s + (1.61 + 2.09i)7-s + (−4.11 + 4.11i)8-s + (−0.866 − 0.499i)9-s + (−0.532 + 5.59i)10-s + (2.73 + 4.74i)11-s + (1.11 + 4.17i)12-s + (−0.579 − 0.579i)13-s + (−5.29 − 4.02i)14-s + (1.82 + 1.29i)15-s + (3.00 − 5.19i)16-s + (4.58 + 1.22i)17-s + ⋯
L(s)  = 1  + (−1.71 + 0.459i)2-s + (−0.149 + 0.557i)3-s + (1.86 − 1.07i)4-s + (0.349 − 0.937i)5-s − 1.02i·6-s + (0.611 + 0.791i)7-s + (−1.45 + 1.45i)8-s + (−0.288 − 0.166i)9-s + (−0.168 + 1.76i)10-s + (0.825 + 1.42i)11-s + (0.322 + 1.20i)12-s + (−0.160 − 0.160i)13-s + (−1.41 − 1.07i)14-s + (0.470 + 0.334i)15-s + (0.750 − 1.29i)16-s + (1.11 + 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.548 - 0.836i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.548 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467095 + 0.252216i\)
\(L(\frac12)\) \(\approx\) \(0.467095 + 0.252216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-0.780 + 2.09i)T \)
7 \( 1 + (-1.61 - 2.09i)T \)
good2 \( 1 + (2.42 - 0.650i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-2.73 - 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.579 + 0.579i)T + 13iT^{2} \)
17 \( 1 + (-4.58 - 1.22i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.220 + 0.381i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.457 + 1.70i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.853iT - 29T^{2} \)
31 \( 1 + (-2.32 + 1.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.249 - 0.0668i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.321iT - 41T^{2} \)
43 \( 1 + (0.631 - 0.631i)T - 43iT^{2} \)
47 \( 1 + (2.12 + 7.91i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (11.0 + 2.96i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.89 + 5.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.73 + 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.38 - 5.16i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + (-2.28 + 8.53i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.02 + 5.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.47 - 8.47i)T + 83iT^{2} \)
89 \( 1 + (-4.03 + 6.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.99 + 5.99i)T - 97iT^{2} \)
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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

−14.51666294661131643908838791227, −12.45332873510793559701067785401, −11.63272078341567021806297086446, −10.19045601112168477039368087954, −9.517841535422293421824636258661, −8.703091899437832397703106982850, −7.70527056947243526268951814843, −6.17583349737814237319650012703, −4.86037562894082669847494837837, −1.72531322066134310174011813701, 1.31696028501363947897185171489, 3.16078130310567738103708266808, 6.17445004369747850126301870907, 7.30113805572743568539173493690, 8.100573711604829469500650629548, 9.376098418302910459757831505568, 10.47008448335539637886277086366, 11.21461410620135527838139651108, 11.92761822656308019491754669705, 13.73274821726120578028026834898

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