Properties

Label 2-405-45.4-c1-0-0
Degree $2$
Conductor $405$
Sign $-0.642 - 0.766i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (−1.93 + 1.11i)2-s + (1.5 − 2.59i)4-s + (−1.93 − 1.11i)5-s + 2.23i·8-s + 5.00·10-s + (0.499 + 0.866i)16-s + 4.47i·17-s − 4·19-s + (−5.80 + 3.35i)20-s + (7.74 + 4.47i)23-s + (2.5 + 4.33i)25-s + (−4 + 6.92i)31-s + (−5.80 − 3.35i)32-s + (−5.00 − 8.66i)34-s + (7.74 − 4.47i)38-s + ⋯
L(s)  = 1  + (−1.36 + 0.790i)2-s + (0.750 − 1.29i)4-s + (−0.866 − 0.499i)5-s + 0.790i·8-s + 1.58·10-s + (0.124 + 0.216i)16-s + 1.08i·17-s − 0.917·19-s + (−1.29 + 0.750i)20-s + (1.61 + 0.932i)23-s + (0.5 + 0.866i)25-s + (−0.718 + 1.24i)31-s + (−1.02 − 0.592i)32-s + (−0.857 − 1.48i)34-s + (1.25 − 0.725i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163193 + 0.349969i\)
\(L(\frac12)\) \(\approx\) \(0.163193 + 0.349969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.93 + 1.11i)T \)
good2 \( 1 + (1.93 - 1.11i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-7.74 - 4.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.74 - 4.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.4 + 8.94i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 - 84.0i)T^{2} \)
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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

−11.15982730518919094839231234710, −10.60582829174185013620351378802, −9.380647069139086456025101386553, −8.745572233669702602913182713159, −8.004893326395897102334949780281, −7.20951703150063010986622023724, −6.27365804962657593005476144460, −4.96533041134637229405097271559, −3.58117302598872744835150085828, −1.34507427513411275027406755404, 0.43020214201801468131864267480, 2.33042012379504187933894482332, 3.40819861972392036889456005048, 4.86278321638879379767305792393, 6.66135345361517135208058077696, 7.48613015698205031967928444639, 8.369814538226362840518952268733, 9.130515977637142807260696169414, 10.08466650379890213667380514949, 10.96770936579281599699642072273

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