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Properties

Label 2-414-207.113-c1-0-1
Degree $2$
Conductor $414$
Sign $-0.869 + 0.494i$
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  + (−0.998 − 0.0475i)2-s + (−0.397 + 1.68i)3-s + (0.995 + 0.0950i)4-s + (−1.35 + 1.29i)5-s + (0.477 − 1.66i)6-s + (−0.763 + 3.96i)7-s + (−0.989 − 0.142i)8-s + (−2.68 − 1.34i)9-s + (1.41 − 1.22i)10-s + (−2.80 − 1.44i)11-s + (−0.555 + 1.64i)12-s + (1.06 − 0.205i)13-s + (0.951 − 3.92i)14-s + (−1.64 − 2.79i)15-s + (0.981 + 0.189i)16-s + (2.58 − 5.65i)17-s + ⋯
L(s)  = 1  + (−0.706 − 0.0336i)2-s + (−0.229 + 0.973i)3-s + (0.497 + 0.0475i)4-s + (−0.606 + 0.578i)5-s + (0.194 − 0.679i)6-s + (−0.288 + 1.49i)7-s + (−0.349 − 0.0503i)8-s + (−0.894 − 0.446i)9-s + (0.447 − 0.387i)10-s + (−0.845 − 0.435i)11-s + (−0.160 + 0.473i)12-s + (0.295 − 0.0570i)13-s + (0.254 − 1.04i)14-s + (−0.423 − 0.722i)15-s + (0.245 + 0.0473i)16-s + (0.626 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.869 + 0.494i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0810522 - 0.306613i\)
\(L(\frac12)\) \(\approx\) \(0.0810522 - 0.306613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 + 0.0475i)T \)
3 \( 1 + (0.397 - 1.68i)T \)
23 \( 1 + (0.840 - 4.72i)T \)
good5 \( 1 + (1.35 - 1.29i)T + (0.237 - 4.99i)T^{2} \)
7 \( 1 + (0.763 - 3.96i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (2.80 + 1.44i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (-1.06 + 0.205i)T + (12.0 - 4.83i)T^{2} \)
17 \( 1 + (-2.58 + 5.65i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (0.258 - 0.117i)T + (12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.0980 + 1.02i)T + (-28.4 + 5.48i)T^{2} \)
31 \( 1 + (4.02 + 3.16i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (0.0301 + 0.102i)T + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (6.42 + 6.74i)T + (-1.95 + 40.9i)T^{2} \)
43 \( 1 + (0.883 + 1.12i)T + (-10.1 + 41.7i)T^{2} \)
47 \( 1 + (9.58 - 5.53i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.33 - 2.69i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-1.11 - 5.78i)T + (-54.7 + 21.9i)T^{2} \)
61 \( 1 + (-5.16 - 12.9i)T + (-44.1 + 42.0i)T^{2} \)
67 \( 1 + (-0.764 - 1.48i)T + (-38.8 + 54.5i)T^{2} \)
71 \( 1 + (3.34 - 5.20i)T + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (2.82 + 6.18i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (3.84 + 1.33i)T + (62.0 + 48.8i)T^{2} \)
83 \( 1 + (-5.72 - 5.45i)T + (3.94 + 82.9i)T^{2} \)
89 \( 1 + (-0.618 - 4.30i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (4.55 - 1.10i)T + (86.2 - 44.4i)T^{2} \)
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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

−11.57629739573184200931691365002, −10.84111223760156494184407279604, −9.828869813555285626161573281148, −9.166023231509109385751240345476, −8.297092886912626472240888986920, −7.28048978621856351849402791958, −5.88674956699876796270854283674, −5.25603345993289887986475949882, −3.47328549173744106866088631995, −2.68459316943922877264935009036, 0.25932282028241844746528298247, 1.63076861555524900015667880101, 3.47442701549880342859087154128, 4.86183954271445334089210924424, 6.32145573896820802305732304149, 7.09966055490920195482756008381, 8.057060413157521463067078799072, 8.374614166564138580036496582275, 10.00098321416358215105510684610, 10.61841559556994190528862027184

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