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Properties

Label 2-405-135.122-c1-0-0
Degree $2$
Conductor $405$
Sign $-0.356 - 0.934i$
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  + (−0.865 − 0.606i)2-s + (−0.301 − 0.829i)4-s + (−2.13 − 0.660i)5-s + (−1.22 − 2.61i)7-s + (−0.788 + 2.94i)8-s + (1.44 + 1.86i)10-s + (−1.51 − 1.80i)11-s + (3.11 + 4.45i)13-s + (−0.530 + 3.00i)14-s + (1.11 − 0.935i)16-s + (0.716 + 2.67i)17-s + (−5.35 + 3.09i)19-s + (0.0972 + 1.97i)20-s + (0.217 + 2.48i)22-s + (−0.646 − 0.301i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.428i)2-s + (−0.150 − 0.414i)4-s + (−0.955 − 0.295i)5-s + (−0.461 − 0.990i)7-s + (−0.278 + 1.04i)8-s + (0.458 + 0.590i)10-s + (−0.457 − 0.545i)11-s + (0.865 + 1.23i)13-s + (−0.141 + 0.804i)14-s + (0.278 − 0.233i)16-s + (0.173 + 0.648i)17-s + (−1.22 + 0.709i)19-s + (0.0217 + 0.440i)20-s + (0.0463 + 0.530i)22-s + (−0.134 − 0.0628i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00344557 + 0.00500293i\)
\(L(\frac12)\) \(\approx\) \(0.00344557 + 0.00500293i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.660i)T \)
good2 \( 1 + (0.865 + 0.606i)T + (0.684 + 1.87i)T^{2} \)
7 \( 1 + (1.22 + 2.61i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (1.51 + 1.80i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-3.11 - 4.45i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.716 - 2.67i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.35 - 3.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.646 + 0.301i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.623 + 3.53i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.18 - 1.88i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (4.26 - 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.70 + 1.00i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.126 + 1.44i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (9.40 - 4.38i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-7.22 - 7.22i)T + 53iT^{2} \)
59 \( 1 + (6.27 + 5.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.64 - 0.960i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-6.11 + 4.28i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-4.24 - 2.44i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.73 + 1.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.8 - 2.26i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.73 + 3.90i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (3.33 + 5.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (18.2 + 1.59i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.16928320548992278024242341141, −10.72843592028598143506313660317, −9.839383575116599230625615498405, −8.694912967753057221920152402138, −8.223951926302411800069254171875, −6.93936163098327043564630560613, −5.88787338658599805501702817437, −4.43147125479236031632076567684, −3.57377100437859105278717662102, −1.58821530740788017294194537623, 0.00489317427004339501535876248, 2.81487455218515131397311867882, 3.77682392131290148908333029739, 5.22237381656131928467889273293, 6.54190887876599327293413372595, 7.36884269742487605936902392452, 8.331478369422429832370538512587, 8.805736285952121678369097234722, 9.939155845557407474068405976975, 10.90938295587139144689948171140

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