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Properties

Label 2-105-105.53-c1-0-2
Degree $2$
Conductor $105$
Sign $-0.710 - 0.704i$
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  + (−0.391 + 1.46i)2-s + (−0.879 + 1.49i)3-s + (−0.246 − 0.142i)4-s + (1.82 + 1.29i)5-s + (−1.83 − 1.86i)6-s + (−1.17 − 2.36i)7-s + (−1.83 + 1.83i)8-s + (−1.45 − 2.62i)9-s + (−2.60 + 2.15i)10-s + (−0.791 − 0.457i)11-s + (0.429 − 0.243i)12-s + (3.07 + 3.07i)13-s + (3.92 − 0.791i)14-s + (−3.53 + 1.58i)15-s + (−2.24 − 3.88i)16-s + (1.16 − 0.311i)17-s + ⋯
L(s)  = 1  + (−0.276 + 1.03i)2-s + (−0.507 + 0.861i)3-s + (−0.123 − 0.0712i)4-s + (0.815 + 0.578i)5-s + (−0.749 − 0.762i)6-s + (−0.444 − 0.895i)7-s + (−0.648 + 0.648i)8-s + (−0.484 − 0.874i)9-s + (−0.822 + 0.682i)10-s + (−0.238 − 0.137i)11-s + (0.124 − 0.0702i)12-s + (0.854 + 0.854i)13-s + (1.04 − 0.211i)14-s + (−0.912 + 0.409i)15-s + (−0.561 − 0.971i)16-s + (0.281 − 0.0755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.710 - 0.704i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.710 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331298 + 0.804514i\)
\(L(\frac12)\) \(\approx\) \(0.331298 + 0.804514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.879 - 1.49i)T \)
5 \( 1 + (-1.82 - 1.29i)T \)
7 \( 1 + (1.17 + 2.36i)T \)
good2 \( 1 + (0.391 - 1.46i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (0.791 + 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \)
17 \( 1 + (-1.16 + 0.311i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.95 + 3.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 0.505i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.774 + 0.207i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (-2.71 + 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.85 + 10.6i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.94 - 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.533 - 0.924i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (-2.10 + 0.564i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.62 + 1.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.38 - 2.38i)T - 83iT^{2} \)
89 \( 1 + (5.64 + 9.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−14.27465968985452144741059649516, −13.61571878366485080399089341272, −11.79821094049667137042577187372, −10.81741786145503840120653668215, −9.828015966412588325421936373741, −8.833165568335685518595688298254, −7.14220743513215475832518816371, −6.38046539980900763529485975758, −5.25421233589555496593630846364, −3.36232665685655901241319687431, 1.38870816652073950795454864894, 2.87084453831115097566146032049, 5.55437632177330768590485965859, 6.22711759228484448306377298645, 8.038144938338322050800212712427, 9.344510785224519598349934695339, 10.26746456174484044721470634627, 11.42643746516770855696824855715, 12.39553590504107012548461111668, 12.86768900673364318847468579194

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