Properties

Label 2-414-23.13-c1-0-2
Degree $2$
Conductor $414$
Sign $-0.651 - 0.758i$
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 + (−0.985 + 2.15i)5-s + (0.381 + 0.112i)7-s + (−0.841 + 0.540i)8-s + (−2.27 + 0.668i)10-s + (−2.10 + 2.42i)11-s + (−0.149 + 0.0437i)13-s + (0.165 + 0.361i)14-s + (−0.959 − 0.281i)16-s + (0.467 + 3.24i)17-s + (−0.404 + 2.81i)19-s + (−1.99 − 1.28i)20-s − 3.21·22-s + (−1.27 − 4.62i)23-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (−0.0711 + 0.494i)4-s + (−0.440 + 0.965i)5-s + (0.144 + 0.0423i)7-s + (−0.297 + 0.191i)8-s + (−0.719 + 0.211i)10-s + (−0.633 + 0.731i)11-s + (−0.0413 + 0.0121i)13-s + (0.0441 + 0.0967i)14-s + (−0.239 − 0.0704i)16-s + (0.113 + 0.788i)17-s + (−0.0929 + 0.646i)19-s + (−0.446 − 0.286i)20-s − 0.684·22-s + (−0.266 − 0.963i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.651 - 0.758i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585659 + 1.27543i\)
\(L(\frac12)\) \(\approx\) \(0.585659 + 1.27543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 4.62i)T \)
good5 \( 1 + (0.985 - 2.15i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.381 - 0.112i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (2.10 - 2.42i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.149 - 0.0437i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.467 - 3.24i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.404 - 2.81i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.0538 - 0.374i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.31 + 1.48i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-2.66 - 5.84i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-1.66 + 3.64i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-6.25 - 4.01i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 2.97T + 47T^{2} \)
53 \( 1 + (-12.5 - 3.67i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-8.29 + 2.43i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-9.37 + 6.02i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (5.50 + 6.35i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-0.233 - 0.269i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.802 - 5.57i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-7.23 + 2.12i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (5.56 + 12.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (1.81 + 1.16i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.09 - 2.39i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.61721067630037638686708448662, −10.64506793253035140556004144562, −9.923525198879067093095823052488, −8.468679450922295807185208419936, −7.72206899540026154522906484518, −6.86278224166289608856293865526, −5.96775759164815168941169532141, −4.72395735772089593677125684258, −3.66849962395502961088759134352, −2.41531730421639251397119420669, 0.798553922923059855003832899893, 2.61929204974270821650352683533, 3.94255919635798712867616102397, 4.96293492057836158028745493498, 5.73841266652632802135737852378, 7.21471071688706583456475656048, 8.275375890785820253457334560123, 9.088793791354665113903372191752, 10.07707362281146085435985073690, 11.16706979859136895113232714736

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