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Properties

Label 2-810-135.122-c1-0-1
Degree $2$
Conductor $810$
Sign $-0.709 - 0.704i$
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  + (0.819 + 0.573i)2-s + (0.342 + 0.939i)4-s + (−1.00 + 1.99i)5-s + (−0.0561 − 0.120i)7-s + (−0.258 + 0.965i)8-s + (−1.96 + 1.06i)10-s + (1.11 + 1.32i)11-s + (−1.22 − 1.74i)13-s + (0.0230 − 0.130i)14-s + (−0.766 + 0.642i)16-s + (1.68 + 6.27i)17-s + (−6.47 + 3.73i)19-s + (−2.22 − 0.258i)20-s + (0.150 + 1.72i)22-s + (1.60 + 0.746i)23-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.448 + 0.893i)5-s + (−0.0212 − 0.0455i)7-s + (−0.0915 + 0.341i)8-s + (−0.622 + 0.336i)10-s + (0.335 + 0.399i)11-s + (−0.339 − 0.484i)13-s + (0.00616 − 0.0349i)14-s + (−0.191 + 0.160i)16-s + (0.407 + 1.52i)17-s + (−1.48 + 0.856i)19-s + (−0.496 − 0.0577i)20-s + (0.0321 + 0.367i)22-s + (0.333 + 0.155i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614444 + 1.49039i\)
\(L(\frac12)\) \(\approx\) \(0.614444 + 1.49039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 \)
5 \( 1 + (1.00 - 1.99i)T \)
good7 \( 1 + (0.0561 + 0.120i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-1.11 - 1.32i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.22 + 1.74i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.68 - 6.27i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.47 - 3.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.60 - 0.746i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.637 - 3.61i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-6.03 + 2.19i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (9.06 - 2.42i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.12 + 0.903i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.171 - 1.96i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (2.27 - 1.06i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (4.81 + 4.81i)T + 53iT^{2} \)
59 \( 1 + (-2.47 - 2.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-10.3 - 3.77i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-7.08 + 4.95i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-8.58 - 4.95i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.62 - 2.57i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0585 - 0.0103i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.58 - 3.69i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-7.54 - 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.11 + 0.0974i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.49467636483998318465749025149, −10.03652296805338761272974596438, −8.463567781074923585121773909097, −8.024045271074348240718184600685, −6.86135738091282930765138569633, −6.40416428166630936240033408798, −5.29414277555178047611715529443, −4.07997402407032382950546901162, −3.41624597376906008730542933910, −2.06123941495302619405335492424, 0.65049859330236411610167442463, 2.24490137920949271093691712570, 3.50647515330740078913627922000, 4.60215033117456886368272931045, 5.10672221151324242270373083193, 6.38195250825429579718352691187, 7.22144273959116591633451698445, 8.429981509305657816985689296363, 9.080083753599006692108842927314, 9.936778361453192827310406249979

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