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Properties

Label 2-408-17.8-c1-0-3
Degree $2$
Conductor $408$
Sign $0.827 + 0.560i$
Analytic cond. $3.25789$
Root an. cond. $1.80496$
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  + (0.382 − 0.923i)3-s + (0.504 + 0.209i)5-s + (2.34 − 0.970i)7-s + (−0.707 − 0.707i)9-s + (1.94 + 4.68i)11-s − 2.53i·13-s + (0.386 − 0.386i)15-s + (1.13 − 3.96i)17-s + (1.03 − 1.03i)19-s − 2.53i·21-s + (−0.313 − 0.756i)23-s + (−3.32 − 3.32i)25-s + (−0.923 + 0.382i)27-s + (7.59 + 3.14i)29-s + (−1.98 + 4.78i)31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (0.225 + 0.0935i)5-s + (0.885 − 0.366i)7-s + (−0.235 − 0.235i)9-s + (0.585 + 1.41i)11-s − 0.701i·13-s + (0.0997 − 0.0997i)15-s + (0.275 − 0.961i)17-s + (0.236 − 0.236i)19-s − 0.553i·21-s + (−0.0653 − 0.157i)23-s + (−0.664 − 0.664i)25-s + (−0.177 + 0.0736i)27-s + (1.40 + 0.583i)29-s + (−0.356 + 0.860i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(408\)    =    \(2^{3} \cdot 3 \cdot 17\)
Sign: $0.827 + 0.560i$
Analytic conductor: \(3.25789\)
Root analytic conductor: \(1.80496\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{408} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 408,\ (\ :1/2),\ 0.827 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61191 - 0.494590i\)
\(L(\frac12)\) \(\approx\) \(1.61191 - 0.494590i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-1.13 + 3.96i)T \)
good5 \( 1 + (-0.504 - 0.209i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.34 + 0.970i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.94 - 4.68i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.53iT - 13T^{2} \)
19 \( 1 + (-1.03 + 1.03i)T - 19iT^{2} \)
23 \( 1 + (0.313 + 0.756i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-7.59 - 3.14i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.98 - 4.78i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.31 + 7.99i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (4.17 - 1.72i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.55 + 1.55i)T + 43iT^{2} \)
47 \( 1 - 4.86iT - 47T^{2} \)
53 \( 1 + (4.85 - 4.85i)T - 53iT^{2} \)
59 \( 1 + (2.24 + 2.24i)T + 59iT^{2} \)
61 \( 1 + (1.20 - 0.499i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 8.60T + 67T^{2} \)
71 \( 1 + (4.49 - 10.8i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (10.8 + 4.49i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-5.33 - 12.8i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (8.06 - 8.06i)T - 83iT^{2} \)
89 \( 1 - 6.97iT - 89T^{2} \)
97 \( 1 + (17.9 + 7.42i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.22155367311665998196406395947, −10.20566356842039546792248607272, −9.378279085432235497290092188822, −8.242343178618825527710364218349, −7.42098576092720900254318183876, −6.67655825589712174843148714150, −5.28496771550577776440444091096, −4.31624319161032436048753764445, −2.70841805580405019798952780282, −1.37047026824774404494423953420, 1.66982574259377548604599095102, 3.30433444334661212253457064974, 4.39846721619229819637838508079, 5.56257265696416946861006529133, 6.39780560877118236190575192729, 7.995678050089038469040620433195, 8.542635842539752727043698718146, 9.462546760264792207243883086796, 10.40853915990372855942527013711, 11.48808742128101204235693761732

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