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Properties

Label 2-810-135.113-c1-0-12
Degree $2$
Conductor $810$
Sign $0.990 - 0.139i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (1.66 − 1.49i)5-s + (4.20 + 2.94i)7-s + (0.965 + 0.258i)8-s + (1.78 − 1.34i)10-s + (0.584 − 1.60i)11-s + (0.125 + 1.43i)13-s + (3.93 + 3.29i)14-s + (0.939 + 0.342i)16-s + (−7.34 + 1.96i)17-s + (−1.39 + 0.805i)19-s + (1.89 − 1.18i)20-s + (0.722 − 1.54i)22-s + (−1.93 − 2.76i)23-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.744 − 0.667i)5-s + (1.58 + 1.11i)7-s + (0.341 + 0.0915i)8-s + (0.565 − 0.424i)10-s + (0.176 − 0.484i)11-s + (0.0348 + 0.397i)13-s + (1.05 + 0.881i)14-s + (0.234 + 0.0855i)16-s + (−1.78 + 0.477i)17-s + (−0.320 + 0.184i)19-s + (0.424 − 0.264i)20-s + (0.153 − 0.330i)22-s + (−0.403 − 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01459 + 0.211586i\)
\(L(\frac12)\) \(\approx\) \(3.01459 + 0.211586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 \)
5 \( 1 + (-1.66 + 1.49i)T \)
good7 \( 1 + (-4.20 - 2.94i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.584 + 1.60i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.125 - 1.43i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (7.34 - 1.96i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.39 - 0.805i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.93 + 2.76i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (2.03 - 1.70i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.579 + 3.28i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.31 - 4.91i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.63 + 7.91i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.762 + 1.63i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (2.07 - 2.96i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (1.69 - 1.69i)T - 53iT^{2} \)
59 \( 1 + (-6.61 + 2.40i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.840 + 4.76i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (10.5 - 0.919i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (3.27 + 1.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.12 + 7.91i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.56 + 11.3i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.331 + 3.78i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (-0.871 - 1.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.401 + 0.187i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.52112521818771626899073862919, −9.060129829582632803052234290827, −8.714533582027847113489718340203, −7.83452458029707648408252763010, −6.39807529321174190525482760140, −5.80151756765409421060961743970, −4.84798934675307930980056597719, −4.25923787450405216739203218337, −2.39278987068796149497295421686, −1.73629104970419557181705367738, 1.56163364722370831429631962200, 2.53848659217167068863436808145, 4.06062369744500193577179360076, 4.71012576765461355489503809228, 5.71524462830955501174698016784, 6.84937709621975333081776323711, 7.36194866741187553403543653021, 8.395837324782662590313996675117, 9.600627599864977298545493936955, 10.49767266385687127755917689999

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