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Properties

Label 2-405-9.4-c1-0-13
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: $\chi_{405} (271, \cdot )$
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

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

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