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Properties

Label 2-105-105.44-c2-0-26
Degree $2$
Conductor $105$
Sign $-0.989 + 0.146i$
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  + (1.48 − 2.57i)2-s + (−1.79 − 2.40i)3-s + (−2.43 − 4.22i)4-s + (−3.23 − 3.80i)5-s + (−8.87 + 1.06i)6-s + (5.16 + 4.72i)7-s − 2.60·8-s + (−2.53 + 8.63i)9-s + (−14.6 + 2.68i)10-s + (−3.56 + 2.05i)11-s + (−5.75 + 13.4i)12-s − 21.3i·13-s + (19.8 − 6.30i)14-s + (−3.32 + 14.6i)15-s + (5.87 − 10.1i)16-s + (1.11 + 1.92i)17-s + ⋯
L(s)  = 1  + (0.744 − 1.28i)2-s + (−0.599 − 0.800i)3-s + (−0.609 − 1.05i)4-s + (−0.647 − 0.761i)5-s + (−1.47 + 0.176i)6-s + (0.738 + 0.674i)7-s − 0.325·8-s + (−0.281 + 0.959i)9-s + (−1.46 + 0.268i)10-s + (−0.323 + 0.186i)11-s + (−0.479 + 1.11i)12-s − 1.64i·13-s + (1.41 − 0.450i)14-s + (−0.221 + 0.975i)15-s + (0.367 − 0.635i)16-s + (0.0653 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.109790 - 1.48960i\)
\(L(\frac12)\) \(\approx\) \(0.109790 - 1.48960i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.79 + 2.40i)T \)
5 \( 1 + (3.23 + 3.80i)T \)
7 \( 1 + (-5.16 - 4.72i)T \)
good2 \( 1 + (-1.48 + 2.57i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (3.56 - 2.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (-1.11 - 1.92i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.93 + 8.55i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.66 + 16.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 28.1iT - 841T^{2} \)
31 \( 1 + (-17.6 - 30.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-43.1 - 24.9i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 27.4iT - 1.68e3T^{2} \)
43 \( 1 + 41.9iT - 1.84e3T^{2} \)
47 \( 1 + (19.1 - 33.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-25.6 - 44.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-22.0 + 12.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.1 - 59.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-59.1 + 34.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 11.9iT - 5.04e3T^{2} \)
73 \( 1 + (88.6 - 51.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-32.1 + 55.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 122.T + 6.88e3T^{2} \)
89 \( 1 + (-116. - 67.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 13.6iT - 9.40e3T^{2} \)
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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

−12.62188429176543034758636366080, −12.22143803587962925307108151871, −11.24886786923054780358080296291, −10.45268037358876074268260773379, −8.559513105344879870623651014746, −7.55529609885061713888156305304, −5.47266329689364508350105926224, −4.76349425342028937981427854928, −2.79490619265118629517914206702, −1.07440162696704758806588195137, 3.89582835566798559926186319547, 4.66043885518469251328931631185, 6.06847833375334732573277113319, 7.08569245479341663557123681823, 8.092993992637788550907315660998, 9.807338245713085225655631679219, 11.12774368309150977021652462925, 11.69010874451277854180289454170, 13.45822235443613584012598511926, 14.43417205947112842229622651175

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