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Properties

Label 2-414-23.9-c1-0-8
Degree $2$
Conductor $414$
Sign $-0.509 + 0.860i$
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 + (−0.459 − 3.19i)5-s + (−0.497 + 1.08i)7-s + (−0.654 + 0.755i)8-s + (1.34 + 2.93i)10-s + (0.544 + 0.159i)11-s + (−2.44 − 5.36i)13-s + (0.170 − 1.18i)14-s + (0.415 − 0.909i)16-s + (0.127 + 0.0817i)17-s + (−3.91 + 2.51i)19-s + (−2.11 − 2.44i)20-s − 0.567·22-s + (−4.67 − 1.07i)23-s + ⋯
L(s)  = 1  + (−0.678 + 0.199i)2-s + (0.420 − 0.270i)4-s + (−0.205 − 1.42i)5-s + (−0.187 + 0.411i)7-s + (−0.231 + 0.267i)8-s + (0.424 + 0.928i)10-s + (0.164 + 0.0481i)11-s + (−0.678 − 1.48i)13-s + (0.0455 − 0.316i)14-s + (0.103 − 0.227i)16-s + (0.0308 + 0.0198i)17-s + (−0.899 + 0.577i)19-s + (−0.472 − 0.545i)20-s − 0.120·22-s + (−0.974 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.509 + 0.860i$
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: \((2,\ 414,\ (\ :1/2),\ -0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307434 - 0.539172i\)
\(L(\frac12)\) \(\approx\) \(0.307434 - 0.539172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.67 + 1.07i)T \)
good5 \( 1 + (0.459 + 3.19i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.497 - 1.08i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.544 - 0.159i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (2.44 + 5.36i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.127 - 0.0817i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (3.91 - 2.51i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (3.56 + 2.29i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-2.64 + 3.05i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-1.33 + 9.31i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.673 + 4.68i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.29 + 2.64i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 + (-1.21 + 2.66i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-0.411 - 0.901i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.33 - 1.53i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-1.25 + 0.368i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-1.48 + 0.434i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.24 - 2.08i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.52 - 7.71i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.787 + 5.47i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-11.5 - 13.3i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.420 - 2.92i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.70118426642033869209971388473, −9.850591073993781686431601771220, −9.000070868932292406638713775146, −8.231132576246569630334860843077, −7.56559900241413451474787321986, −6.02740513722493794140288464755, −5.30400127754873327810478377984, −4.02128587607792541308879222493, −2.21524851165701728101401743087, −0.47573873478525227925700340348, 2.06258154981672036545917104079, 3.25998572974785068556396279772, 4.44429199580957863881023896842, 6.38457721154778589344116482511, 6.86589507592683971020053384727, 7.72647039926432980127808669100, 8.910859475372999772601411995951, 9.889804229674853931004416029637, 10.52187271212762295573172233720, 11.42944132843606298357238765305

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