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Properties

Label 2-414-23.16-c1-0-5
Degree $2$
Conductor $414$
Sign $0.761 + 0.647i$
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 + (0.614 + 1.34i)5-s + (3.07 − 0.902i)7-s + (−0.841 − 0.540i)8-s + (1.41 + 0.416i)10-s + (0.0362 + 0.0417i)11-s + (0.773 + 0.227i)13-s + (1.33 − 2.91i)14-s + (−0.959 + 0.281i)16-s + (0.293 − 2.03i)17-s + (0.523 + 3.63i)19-s + (1.24 − 0.799i)20-s + 0.0552·22-s + (4.62 + 1.25i)23-s + ⋯
L(s)  = 1  + (0.463 − 0.534i)2-s + (−0.0711 − 0.494i)4-s + (0.274 + 0.601i)5-s + (1.16 − 0.341i)7-s + (−0.297 − 0.191i)8-s + (0.448 + 0.131i)10-s + (0.0109 + 0.0125i)11-s + (0.214 + 0.0629i)13-s + (0.355 − 0.778i)14-s + (−0.239 + 0.0704i)16-s + (0.0710 − 0.494i)17-s + (0.120 + 0.834i)19-s + (0.278 − 0.178i)20-s + 0.0117·22-s + (0.965 + 0.261i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.761 + 0.647i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87720 - 0.689940i\)
\(L(\frac12)\) \(\approx\) \(1.87720 - 0.689940i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.62 - 1.25i)T \)
good5 \( 1 + (-0.614 - 1.34i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-3.07 + 0.902i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-0.0362 - 0.0417i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.773 - 0.227i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.293 + 2.03i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.523 - 3.63i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-1.04 + 7.27i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (6.69 + 4.30i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.28 - 2.80i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.606 - 1.32i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (10.5 - 6.77i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 + (10.5 - 3.11i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (5.14 + 1.51i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-10.2 - 6.59i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (2.96 - 3.41i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (9.56 - 11.0i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.40 - 9.76i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (3.47 + 1.01i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-4.16 + 9.12i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-6.94 + 4.46i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-2.91 - 6.38i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.26791833537546097585055903555, −10.39682715594372642445482666284, −9.583149679659640042792958078951, −8.336994145084411150082904750260, −7.38225892467493193193488838520, −6.24709867766326286904762177926, −5.16259516413157436926586688013, −4.16976973857629012910380833485, −2.86857400310562706740346046828, −1.51726844555651435702275133698, 1.67083310345128625646662225527, 3.39130389496039825312668542659, 4.98234532306531659011268304344, 5.16725392528263191533848916957, 6.59245455580441646841214861134, 7.57467264776188172244753062425, 8.682700097019160160323757223917, 9.040128576090070045874354032685, 10.64062384644863690583366501656, 11.34766644933378679675889001008

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