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Properties

Label 2-810-135.113-c1-0-7
Degree $2$
Conductor $810$
Sign $0.767 - 0.641i$
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 + (1.77 + 1.35i)5-s + (−1.16 − 0.813i)7-s + (0.965 + 0.258i)8-s + (1.65 + 1.50i)10-s + (−0.290 + 0.797i)11-s + (0.499 + 5.70i)13-s + (−1.08 − 0.911i)14-s + (0.939 + 0.342i)16-s + (6.26 − 1.67i)17-s + (0.472 − 0.272i)19-s + (1.51 + 1.64i)20-s + (−0.358 + 0.768i)22-s + (−2.49 − 3.56i)23-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.795 + 0.605i)5-s + (−0.439 − 0.307i)7-s + (0.341 + 0.0915i)8-s + (0.523 + 0.475i)10-s + (−0.0874 + 0.240i)11-s + (0.138 + 1.58i)13-s + (−0.290 − 0.243i)14-s + (0.234 + 0.0855i)16-s + (1.51 − 0.407i)17-s + (0.108 − 0.0625i)19-s + (0.339 + 0.367i)20-s + (−0.0764 + 0.163i)22-s + (−0.520 − 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.767 - 0.641i$
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: \((2,\ 810,\ (\ :1/2),\ 0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49462 + 0.905132i\)
\(L(\frac12)\) \(\approx\) \(2.49462 + 0.905132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 - 1.35i)T \)
good7 \( 1 + (1.16 + 0.813i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (0.290 - 0.797i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.499 - 5.70i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-6.26 + 1.67i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.472 + 0.272i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.49 + 3.56i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-2.18 + 1.83i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.43 - 8.12i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.0155 - 0.0581i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.57 + 7.84i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.71 + 7.96i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (4.34 - 6.20i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (3.24 - 3.24i)T - 53iT^{2} \)
59 \( 1 + (-1.79 + 0.652i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.615 + 3.49i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.81 + 0.158i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-4.81 - 2.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.64 - 9.85i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.97 + 10.6i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.723 + 8.27i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (4.25 + 7.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.1 - 6.59i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.27876188515224640202675504619, −9.742028241338284072215974203187, −8.732747505033388484611009299515, −7.37777468837415878755355597501, −6.78582315891911566579554438476, −6.00115200240720579436214348935, −5.03477868752743319500338951275, −3.90364548676372794274270234779, −2.90160474990630983399770271029, −1.71727445756237531854807174919, 1.20368806808034859182551038209, 2.70818813858488435213407244338, 3.60137116631578872685919686146, 4.98622671507758402715019606024, 5.76284297989812337667341436911, 6.17706294159472754740411936336, 7.70359737288234706906414739897, 8.286098835450069762747079827753, 9.671072960852996599390426772281, 9.968774626745706083642586468554

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