↑

Properties

Label 2-414-207.113-c1-0-20
Degree $2$
Conductor $414$
Sign $0.999 - 0.0285i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.998 + 0.0475i)2-s + (1.50 + 0.850i)3-s + (0.995 + 0.0950i)4-s + (0.631 − 0.601i)5-s + (1.46 + 0.921i)6-s + (0.849 − 4.40i)7-s + (0.989 + 0.142i)8-s + (1.55 + 2.56i)9-s + (0.659 − 0.571i)10-s + (0.387 + 0.199i)11-s + (1.42 + 0.989i)12-s + (−5.47 + 1.05i)13-s + (1.05 − 4.36i)14-s + (1.46 − 0.371i)15-s + (0.981 + 0.189i)16-s + (0.134 − 0.295i)17-s + ⋯
L(s)  = 1  + (0.706 + 0.0336i)2-s + (0.871 + 0.490i)3-s + (0.497 + 0.0475i)4-s + (0.282 − 0.269i)5-s + (0.598 + 0.376i)6-s + (0.321 − 1.66i)7-s + (0.349 + 0.0503i)8-s + (0.518 + 0.855i)9-s + (0.208 − 0.180i)10-s + (0.116 + 0.0602i)11-s + (0.410 + 0.285i)12-s + (−1.51 + 0.292i)13-s + (0.282 − 1.16i)14-s + (0.378 − 0.0959i)15-s + (0.245 + 0.0473i)16-s + (0.0326 − 0.0715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.999 - 0.0285i$
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: \((2,\ 414,\ (\ :1/2),\ 0.999 - 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75650 + 0.0393086i\)
\(L(\frac12)\) \(\approx\) \(2.75650 + 0.0393086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 0.850i)T \)
23 \( 1 + (-0.486 + 4.77i)T \)
good5 \( 1 + (-0.631 + 0.601i)T + (0.237 - 4.99i)T^{2} \)
7 \( 1 + (-0.849 + 4.40i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (-0.387 - 0.199i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (5.47 - 1.05i)T + (12.0 - 4.83i)T^{2} \)
17 \( 1 + (-0.134 + 0.295i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (5.74 - 2.62i)T + (12.4 - 14.3i)T^{2} \)
29 \( 1 + (-0.750 - 7.86i)T + (-28.4 + 5.48i)T^{2} \)
31 \( 1 + (-2.09 - 1.64i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (-1.31 - 4.48i)T + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (5.29 + 5.54i)T + (-1.95 + 40.9i)T^{2} \)
43 \( 1 + (-7.50 - 9.53i)T + (-10.1 + 41.7i)T^{2} \)
47 \( 1 + (3.75 - 2.16i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.70 + 6.57i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (0.803 + 4.16i)T + (-54.7 + 21.9i)T^{2} \)
61 \( 1 + (3.56 + 8.89i)T + (-44.1 + 42.0i)T^{2} \)
67 \( 1 + (2.01 + 3.90i)T + (-38.8 + 54.5i)T^{2} \)
71 \( 1 + (-6.80 + 10.5i)T + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (-0.190 - 0.417i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (4.13 + 1.43i)T + (62.0 + 48.8i)T^{2} \)
83 \( 1 + (-0.753 - 0.718i)T + (3.94 + 82.9i)T^{2} \)
89 \( 1 + (1.48 + 10.3i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-0.991 + 0.240i)T + (86.2 - 44.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.98836817179259449049734429050, −10.36777104209645028855658362092, −9.587616813887639877465513399769, −8.365474407351315161300907459136, −7.44404699268230381169818249580, −6.68743270812760000989212628400, −4.89135701370363439749291601927, −4.43827860927224643951922410347, −3.29639846234176971851457099972, −1.83692349592002959530753275485, 2.29531514243024567696800424399, 2.58742685059584877968433635299, 4.26487290764249245095203327953, 5.51185952032919103690489810799, 6.39649761538384003722456568664, 7.48568390320152218870674090100, 8.434512912614777652262482860620, 9.305675168487973549296209316887, 10.22296210038340672737393154669, 11.67067831080648107045466437741

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