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Properties

Label 2-405-45.43-c2-0-4
Degree $2$
Conductor $405$
Sign $0.999 - 0.0256i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 3.50i)2-s + (−7.95 + 4.59i)4-s + (4.78 + 1.44i)5-s + (−1.66 − 6.20i)7-s + (13.3 + 13.3i)8-s + (0.573 − 18.1i)10-s + (−3.66 + 6.34i)11-s + (−5.08 + 18.9i)13-s + (−20.2 + 11.6i)14-s + (15.8 − 27.4i)16-s + (−20.9 + 20.9i)17-s − 0.814i·19-s + (−44.7 + 10.4i)20-s + (25.7 + 6.88i)22-s + (−5.74 + 21.4i)23-s + ⋯
L(s)  = 1  + (−0.469 − 1.75i)2-s + (−1.98 + 1.14i)4-s + (0.957 + 0.289i)5-s + (−0.237 − 0.886i)7-s + (1.66 + 1.66i)8-s + (0.0573 − 1.81i)10-s + (−0.333 + 0.576i)11-s + (−0.391 + 1.45i)13-s + (−1.44 + 0.833i)14-s + (0.989 − 1.71i)16-s + (−1.23 + 1.23i)17-s − 0.0428i·19-s + (−2.23 + 0.524i)20-s + (1.16 + 0.313i)22-s + (−0.249 + 0.931i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.999 - 0.0256i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.999 - 0.0256i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.761007 + 0.00977780i\)
\(L(\frac12)\) \(\approx\) \(0.761007 + 0.00977780i\)
\(L(2)\) 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 \( 1 + (-4.78 - 1.44i)T \)
good2 \( 1 + (0.939 + 3.50i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (1.66 + 6.20i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.66 - 6.34i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (5.08 - 18.9i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (20.9 - 20.9i)T - 289iT^{2} \)
19 \( 1 + 0.814iT - 361T^{2} \)
23 \( 1 + (5.74 - 21.4i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (-19.8 - 11.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.7 + 41.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.0 + 11.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-1.40 - 2.43i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (56.3 - 15.0i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-19.0 - 71.0i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-20.9 - 20.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-33.7 + 19.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-2.86 + 4.96i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (75.0 + 20.1i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 62.1T + 5.04e3T^{2} \)
73 \( 1 + (-8.47 - 8.47i)T + 5.32e3iT^{2} \)
79 \( 1 + (-23.0 - 13.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-70.8 + 18.9i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 79.7iT - 7.92e3T^{2} \)
97 \( 1 + (-22.5 - 84.2i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−10.90243878515856163602502534428, −10.23122340680637045560046247937, −9.535833538606231969907227810889, −8.879522771638536783651641868870, −7.48960109236010985511032364233, −6.35636432234612804502263376468, −4.64738525856726491712207884287, −3.81853454607687186038709270759, −2.36647561519898436598639132449, −1.57453210175292329261496224087, 0.38369436190004330046997816498, 2.66101935898575441109895220977, 4.91612873312957780459872926811, 5.45794003509972204188215701855, 6.30412460057562596790929625024, 7.16987948090349424649875477661, 8.495439840670476088920225715018, 8.760824377186017683550285216906, 9.801927772940675282776924647942, 10.53146084960669398331900941364

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