Properties

Label 2-405-135.92-c1-0-8
Degree $2$
Conductor $405$
Sign $-0.213 + 0.977i$
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  + (−1.98 + 0.174i)2-s + (1.95 − 0.344i)4-s + (−0.840 − 2.07i)5-s + (1.15 − 0.808i)7-s + (0.0258 − 0.00692i)8-s + (2.03 + 3.97i)10-s + (1.21 + 3.35i)11-s + (0.265 − 3.03i)13-s + (−2.15 + 1.80i)14-s + (−3.78 + 1.37i)16-s + (−3.91 − 1.05i)17-s + (−1.42 − 0.821i)19-s + (−2.35 − 3.76i)20-s + (−3.00 − 6.45i)22-s + (3.67 − 5.24i)23-s + ⋯
L(s)  = 1  + (−1.40 + 0.123i)2-s + (0.978 − 0.172i)4-s + (−0.376 − 0.926i)5-s + (0.436 − 0.305i)7-s + (0.00914 − 0.00244i)8-s + (0.642 + 1.25i)10-s + (0.367 + 1.01i)11-s + (0.0737 − 0.842i)13-s + (−0.576 + 0.483i)14-s + (−0.945 + 0.344i)16-s + (−0.950 − 0.254i)17-s + (−0.326 − 0.188i)19-s + (−0.527 − 0.841i)20-s + (−0.641 − 1.37i)22-s + (0.765 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.213 + 0.977i$
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: \((2,\ 405,\ (\ :1/2),\ -0.213 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312837 - 0.388413i\)
\(L(\frac12)\) \(\approx\) \(0.312837 - 0.388413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.840 + 2.07i)T \)
good2 \( 1 + (1.98 - 0.174i)T + (1.96 - 0.347i)T^{2} \)
7 \( 1 + (-1.15 + 0.808i)T + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (-1.21 - 3.35i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.265 + 3.03i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (3.91 + 1.05i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.42 + 0.821i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 5.24i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (-2.12 - 1.78i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.13 + 6.43i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.751 + 2.80i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.92 + 7.06i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.41 + 5.17i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (4.56 + 6.52i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (8.37 + 8.37i)T + 53iT^{2} \)
59 \( 1 + (-13.4 - 4.89i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.11 + 6.32i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (11.6 + 1.01i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (0.966 - 0.557i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.19 - 8.19i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.51 - 4.18i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.529 + 6.05i)T + (-81.7 + 14.4i)T^{2} \)
89 \( 1 + (-0.0742 + 0.128i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.6 - 5.45i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.76721978367913800324480571232, −9.933933691182233739126639937760, −8.985332805518381641657529413106, −8.442210210892784861307891464365, −7.53396456683559541568403424431, −6.73742410202722721357726035707, −5.04102974860516291121768808860, −4.17566641337553446147819584354, −2.00172118187418235699160873386, −0.53542751125153900756630352204, 1.61343878441558886495023409135, 3.08710580731479194471910021264, 4.56342392704262127059344180467, 6.25874561203915649540417388939, 7.03838823916450741844675708087, 8.096325570081498140315859247876, 8.743813929404070663641513606919, 9.599104667530696931972052762489, 10.64556750566002493502778490805, 11.29823325098816548802518915930

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