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Properties

Label 2-414-23.6-c1-0-9
Degree $2$
Conductor $414$
Sign $-0.968 + 0.248i$
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  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−2.43 + 1.56i)5-s + (0.394 − 2.74i)7-s + (−0.959 + 0.281i)8-s + (0.411 + 2.86i)10-s + (−2.26 − 4.95i)11-s + (−0.0520 − 0.361i)13-s + (−2.33 − 1.49i)14-s + (−0.142 + 0.989i)16-s + (−4.12 + 4.75i)17-s + (−4.01 − 4.63i)19-s + (2.77 + 0.815i)20-s − 5.45·22-s + (0.965 − 4.69i)23-s + ⋯
L(s)  = 1  + (0.293 − 0.643i)2-s + (−0.327 − 0.377i)4-s + (−1.08 + 0.699i)5-s + (0.149 − 1.03i)7-s + (−0.339 + 0.0996i)8-s + (0.130 + 0.905i)10-s + (−0.682 − 1.49i)11-s + (−0.0144 − 0.100i)13-s + (−0.622 − 0.400i)14-s + (−0.0355 + 0.247i)16-s + (−1.00 + 1.15i)17-s + (−0.921 − 1.06i)19-s + (0.620 + 0.182i)20-s − 1.16·22-s + (0.201 − 0.979i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0903978 - 0.717231i\)
\(L(\frac12)\) \(\approx\) \(0.0903978 - 0.717231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 \)
23 \( 1 + (-0.965 + 4.69i)T \)
good5 \( 1 + (2.43 - 1.56i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.394 + 2.74i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (2.26 + 4.95i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.0520 + 0.361i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (4.12 - 4.75i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (4.01 + 4.63i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-2.23 + 2.57i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (7.37 - 2.16i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-4.06 - 2.61i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-3.56 + 2.29i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-6.75 - 1.98i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 + (-1.01 + 7.06i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.0626 - 0.435i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (7.80 - 2.29i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-1.84 + 4.04i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-0.586 + 1.28i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-2.58 - 2.97i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.565 - 3.93i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (13.5 + 8.70i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-13.2 - 3.87i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (3.64 - 2.34i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.85840281730990304229200768761, −10.54766849114556983572436792911, −8.873799039145278154072192288671, −8.110004499899822164641073482806, −7.06930220212930537501113498542, −6.05090446899089265754463570471, −4.48358997939006209469128673607, −3.76643251577806236580241218758, −2.64586328112987750442159061409, −0.41019807479622127931905416589, 2.34643332171757693932392622661, 4.08359950865950002198377982291, 4.83177775159997837715305187959, 5.78322391285376048302581366077, 7.22183603861448727413238404133, 7.78209235248272397911835875934, 8.823932565073850642420835954398, 9.450435184479263659167429382970, 10.90541621636792446085178360463, 11.97859138304554325771514536288

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