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Properties

Label 2-414-23.9-c1-0-0
Degree $2$
Conductor $414$
Sign $-0.921 - 0.389i$
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  + (−0.959 + 0.281i)2-s + (0.841 − 0.540i)4-s + (0.592 + 4.11i)5-s + (−1.22 + 2.67i)7-s + (−0.654 + 0.755i)8-s + (−1.72 − 3.78i)10-s + (−3.15 − 0.925i)11-s + (−0.0583 − 0.127i)13-s + (0.418 − 2.91i)14-s + (0.415 − 0.909i)16-s + (−5.26 − 3.38i)17-s + (1.99 − 1.28i)19-s + (2.72 + 3.14i)20-s + 3.28·22-s + (0.435 − 4.77i)23-s + ⋯
L(s)  = 1  + (−0.678 + 0.199i)2-s + (0.420 − 0.270i)4-s + (0.264 + 1.84i)5-s + (−0.462 + 1.01i)7-s + (−0.231 + 0.267i)8-s + (−0.546 − 1.19i)10-s + (−0.950 − 0.278i)11-s + (−0.0161 − 0.0354i)13-s + (0.111 − 0.778i)14-s + (0.103 − 0.227i)16-s + (−1.27 − 0.819i)17-s + (0.458 − 0.294i)19-s + (0.609 + 0.703i)20-s + 0.700·22-s + (0.0908 − 0.995i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.921 - 0.389i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134227 + 0.662688i\)
\(L(\frac12)\) \(\approx\) \(0.134227 + 0.662688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 \)
23 \( 1 + (-0.435 + 4.77i)T \)
good5 \( 1 + (-0.592 - 4.11i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (1.22 - 2.67i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (3.15 + 0.925i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.0583 + 0.127i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (5.26 + 3.38i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-1.99 + 1.28i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-8.36 - 5.37i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (1.49 - 1.72i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.883 - 6.14i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.918 - 6.38i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.839 - 0.968i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 2.84T + 47T^{2} \)
53 \( 1 + (1.80 - 3.95i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (0.0304 + 0.0666i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.25 - 1.45i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (4.15 - 1.22i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-9.39 + 2.75i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-4.61 + 2.96i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.99 - 8.75i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.56 - 10.8i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-8.58 - 9.90i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.73 - 12.0i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.23349212783858974260948825755, −10.67926588173631159684138159234, −9.870255549765396272835956130436, −8.956895859651375525426960998915, −7.914310792403775242997185627657, −6.74430919897945957649738960752, −6.41570089539618870336889336655, −5.11841863393256346432815966217, −2.92753112966237397742877790696, −2.55765794053775395468614762605, 0.51564030423045839399844368328, 1.95876949515390311985141609466, 3.88708179551608018045836703866, 4.88499722550296161387993204827, 6.05933251798835906359634559662, 7.39511292202204586264286188991, 8.216728249817943521581295490946, 9.041119856496148296535593181689, 9.855597157278515579455641719731, 10.56977147240435806163388431930

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