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Properties

Label 2-414-23.16-c1-0-7
Degree $2$
Conductor $414$
Sign $0.218 + 0.975i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (−0.654 + 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.810 − 1.77i)5-s + (0.439 − 0.129i)7-s + (0.841 + 0.540i)8-s + (1.87 + 0.549i)10-s + (0.824 + 0.951i)11-s + (−5.37 − 1.57i)13-s + (−0.190 + 0.416i)14-s + (−0.959 + 0.281i)16-s + (0.931 − 6.47i)17-s + (−0.301 − 2.09i)19-s + (−1.64 + 1.05i)20-s − 1.25·22-s + (−2.94 − 3.78i)23-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (−0.0711 − 0.494i)4-s + (−0.362 − 0.794i)5-s + (0.166 − 0.0487i)7-s + (0.297 + 0.191i)8-s + (0.592 + 0.173i)10-s + (0.248 + 0.287i)11-s + (−1.49 − 0.437i)13-s + (−0.0508 + 0.111i)14-s + (−0.239 + 0.0704i)16-s + (0.225 − 1.57i)17-s + (−0.0692 − 0.481i)19-s + (−0.367 + 0.235i)20-s − 0.268·22-s + (−0.613 − 0.789i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588542 - 0.471439i\)
\(L(\frac12)\) \(\approx\) \(0.588542 - 0.471439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 \)
23 \( 1 + (2.94 + 3.78i)T \)
good5 \( 1 + (0.810 + 1.77i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-0.439 + 0.129i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-0.824 - 0.951i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (5.37 + 1.57i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.931 + 6.47i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (0.301 + 2.09i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.720 + 5.01i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (0.0916 + 0.0589i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (0.384 - 0.842i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-4.08 - 8.95i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.69 + 3.66i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 2.50T + 47T^{2} \)
53 \( 1 + (2.53 - 0.745i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (4.10 + 1.20i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (7.22 + 4.64i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-9.84 + 11.3i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (0.198 - 0.229i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.476 + 3.31i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-7.72 - 2.26i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (4.35 - 9.54i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (5.01 - 3.22i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-7.58 - 16.6i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.95993505313867873689269501140, −9.753658027437366076727432682309, −9.307293007040754423324032936246, −8.099351474507093810755884257886, −7.52589204966771208239765164584, −6.45040671058987845709047922671, −5.06198704821813767990465510735, −4.52600634427105208712256054665, −2.55048765223663611210950938795, −0.56251735770526782435790267981, 1.86198345000859141576885888319, 3.21542685522273506641092543395, 4.24433632365052394615846440897, 5.74805861047475757821947166681, 7.01423259281156396878759209903, 7.73018736758284111248135989596, 8.750443859739429038245022374531, 9.773160620254748162300130137955, 10.54463977939934895595908979424, 11.28152815397398852789365541840

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