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Properties

Label 2-405-9.7-c1-0-6
Degree $2$
Conductor $405$
Sign $0.939 - 0.342i$
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  + (0.651 − 1.12i)2-s + (0.151 + 0.262i)4-s + (0.5 + 0.866i)5-s + (−2.30 + 3.98i)7-s + 3·8-s + 1.30·10-s + (−1.30 + 2.25i)11-s + (0.302 + 0.524i)13-s + (2.99 + 5.19i)14-s + (1.65 − 2.86i)16-s + 5.60·17-s − 3.60·19-s + (−0.151 + 0.262i)20-s + (1.69 + 2.93i)22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.460 − 0.797i)2-s + (0.0756 + 0.131i)4-s + (0.223 + 0.387i)5-s + (−0.870 + 1.50i)7-s + 1.06·8-s + 0.411·10-s + (−0.392 + 0.680i)11-s + (0.0839 + 0.145i)13-s + (0.801 + 1.38i)14-s + (0.412 − 0.715i)16-s + 1.35·17-s − 0.827·19-s + (−0.0338 + 0.0586i)20-s + (0.361 + 0.626i)22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76225 + 0.310732i\)
\(L(\frac12)\) \(\approx\) \(1.76225 + 0.310732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.651 + 1.12i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (2.30 - 3.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.30 - 2.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.302 - 0.524i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.30 + 7.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.802 + 1.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.30 - 2.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.30 + 5.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.60T + 53T^{2} \)
59 \( 1 + (4.30 + 7.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.10 - 8.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.60 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 5.39T + 73T^{2} \)
79 \( 1 + (-2.19 + 3.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.81T + 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59709406586318366744744879716, −10.34841825326314720597237303494, −9.842338608551159867765935698660, −8.650636348147380078291939066124, −7.60040465057435819924002198578, −6.41305659278662071023006572924, −5.49255452988697611010933655776, −4.12983989276700198243143950692, −2.86433714449689844625733203900, −2.21165622927383267628509703333, 1.09931233469410282585184932909, 3.32602525243438630450987156500, 4.45468063577354559827979660202, 5.59092157352171441693458352360, 6.40157761051779049411797018760, 7.32436297571220941291849413240, 8.095683912958206671642175492552, 9.500577736619976487005879500664, 10.43417004346939799285746377735, 10.84046492244913769862579747778

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