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Properties

Label 2-405-45.7-c2-0-11
Degree $2$
Conductor $405$
Sign $-0.176 - 0.984i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
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  + (2.15 − 0.578i)2-s + (0.866 − 0.5i)4-s + (−3.31 + 3.74i)5-s + (6.83 − 1.83i)7-s + (−4.74 + 4.74i)8-s + (−5 + 10i)10-s + (−7.90 + 13.6i)11-s + (−13.6 − 3.66i)13-s + (13.6 − 7.90i)14-s + (−9.49 + 16.4i)16-s + (3.16 + 3.16i)17-s + 18i·19-s + (−1.00 + 4.89i)20-s + (−9.15 + 34.1i)22-s + (−4.31 − 1.15i)23-s + ⋯
L(s)  = 1  + (1.07 − 0.289i)2-s + (0.216 − 0.125i)4-s + (−0.663 + 0.748i)5-s + (0.975 − 0.261i)7-s + (−0.592 + 0.592i)8-s + (−0.5 + i)10-s + (−0.718 + 1.24i)11-s + (−1.05 − 0.281i)13-s + (0.978 − 0.564i)14-s + (−0.593 + 1.02i)16-s + (0.186 + 0.186i)17-s + 0.947i·19-s + (−0.0501 + 0.244i)20-s + (−0.415 + 1.55i)22-s + (−0.187 − 0.0503i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.176 - 0.984i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.176 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17472 + 1.40377i\)
\(L(\frac12)\) \(\approx\) \(1.17472 + 1.40377i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (3.31 - 3.74i)T \)
good2 \( 1 + (-2.15 + 0.578i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (-6.83 + 1.83i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (7.90 - 13.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (13.6 + 3.66i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (-3.16 - 3.16i)T + 289iT^{2} \)
19 \( 1 - 18iT - 361T^{2} \)
23 \( 1 + (4.31 + 1.15i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-41.0 - 23.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-10 - 10i)T + 1.36e3iT^{2} \)
41 \( 1 + (-15.8 - 27.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.66 - 13.6i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (56.1 - 15.0i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (25.2 - 25.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (-41.0 + 23.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29 + 50.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-25.6 + 95.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 63.2T + 5.04e3T^{2} \)
73 \( 1 + (-55 + 55i)T - 5.32e3iT^{2} \)
79 \( 1 + (10.3 + 6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (19.6 + 73.4i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (-6.83 + 1.83i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.49945837824783245066322110258, −10.62165611856763363431691020196, −9.783415483716555211529616640188, −8.117588985210602963336378824402, −7.70251528290692197337721884145, −6.46370384164709270136836121304, −5.00479386010629767371990815981, −4.54641298182923400719600449977, −3.30051113190840078345788710464, −2.17756299101604149336355291148, 0.54715903863425491032257209176, 2.73108853449954190467969906399, 4.10581530603600100404992290225, 4.99501916716361697928433680631, 5.50170623691002039660780574756, 6.89770848509319085774816139785, 8.042857356724882133434395128832, 8.721963373734073360052043065788, 9.834706008646479925558087178881, 11.22472552585445105783883174010

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