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Properties

Label 2-405-135.92-c1-0-12
Degree $2$
Conductor $405$
Sign $0.998 - 0.0553i$
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  + (2.71 − 0.237i)2-s + (5.34 − 0.942i)4-s + (−1.08 + 1.95i)5-s + (−1.29 + 0.907i)7-s + (9.01 − 2.41i)8-s + (−2.47 + 5.56i)10-s + (−0.162 − 0.447i)11-s + (0.242 − 2.76i)13-s + (−3.30 + 2.77i)14-s + (13.7 − 4.99i)16-s + (−3.35 − 0.898i)17-s + (−1.97 − 1.13i)19-s + (−3.93 + 11.4i)20-s + (−0.548 − 1.17i)22-s + (0.230 − 0.329i)23-s + ⋯
L(s)  = 1  + (1.91 − 0.167i)2-s + (2.67 − 0.471i)4-s + (−0.484 + 0.875i)5-s + (−0.489 + 0.343i)7-s + (3.18 − 0.854i)8-s + (−0.782 + 1.76i)10-s + (−0.0491 − 0.134i)11-s + (0.0671 − 0.767i)13-s + (−0.882 + 0.740i)14-s + (3.42 − 1.24i)16-s + (−0.813 − 0.217i)17-s + (−0.452 − 0.261i)19-s + (−0.881 + 2.56i)20-s + (−0.116 − 0.250i)22-s + (0.0481 − 0.0687i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.998 - 0.0553i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.998 - 0.0553i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.71206 + 0.102794i\)
\(L(\frac12)\) \(\approx\) \(3.71206 + 0.102794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.95i)T \)
good2 \( 1 + (-2.71 + 0.237i)T + (1.96 - 0.347i)T^{2} \)
7 \( 1 + (1.29 - 0.907i)T + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (0.162 + 0.447i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.242 + 2.76i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (3.35 + 0.898i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.97 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.230 + 0.329i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (4.29 + 3.60i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.908 - 5.15i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.387 - 1.44i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.99 - 5.95i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.501 - 1.07i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (6.94 + 9.92i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-0.274 - 0.274i)T + 53iT^{2} \)
59 \( 1 + (-3.33 - 1.21i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.59 - 9.05i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-13.1 - 1.15i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (-13.8 + 7.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.07 - 7.73i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.85 - 3.40i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.253 + 2.90i)T + (-81.7 + 14.4i)T^{2} \)
89 \( 1 + (3.50 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.86 + 3.19i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.37430883566871262394242202664, −10.89687368623936235792091310032, −9.897827103267709381096840327343, −8.136954952399243434868284187795, −6.98216868854843341750126657733, −6.37803980585457090802273207954, −5.38606274676468877443701932755, −4.22168149056728636552016943852, −3.24139320225997801626005584264, −2.41454747057818454791051454506, 2.02943676754998096740393303618, 3.63798826640265584363839366835, 4.28536303503690167410794378128, 5.22214013936771089574236916134, 6.30513770840002496538881777620, 7.12038710423818337374988410894, 8.148768312047675898540628507857, 9.480223398857771720615185814559, 10.93485915479049658171557551990, 11.48573438072063975028676633312

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