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Properties

Label 2-1224-17.8-c1-0-6
Degree $2$
Conductor $1224$
Sign $0.996 + 0.0863i$
Analytic cond. $9.77368$
Root an. cond. $3.12629$
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  + (−3.31 − 1.37i)5-s + (−3.66 + 1.51i)7-s + (0.147 + 0.355i)11-s + 1.23i·13-s + (4.06 + 0.666i)17-s + (−0.0317 + 0.0317i)19-s + (−2.04 − 4.93i)23-s + (5.57 + 5.57i)25-s + (4.62 + 1.91i)29-s + (0.946 − 2.28i)31-s + 14.2·35-s + (1.90 − 4.59i)37-s + (−1.75 + 0.728i)41-s + (6.76 + 6.76i)43-s − 2.73i·47-s + ⋯
L(s)  = 1  + (−1.48 − 0.614i)5-s + (−1.38 + 0.574i)7-s + (0.0444 + 0.107i)11-s + 0.342i·13-s + (0.986 + 0.161i)17-s + (−0.00727 + 0.00727i)19-s + (−0.426 − 1.02i)23-s + (1.11 + 1.11i)25-s + (0.858 + 0.355i)29-s + (0.170 − 0.410i)31-s + 2.40·35-s + (0.312 − 0.755i)37-s + (−0.274 + 0.113i)41-s + (1.03 + 1.03i)43-s − 0.398i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.996 + 0.0863i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.996 + 0.0863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8669774929\)
\(L(\frac12)\) \(\approx\) \(0.8669774929\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + (-4.06 - 0.666i)T \)
good5 \( 1 + (3.31 + 1.37i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.66 - 1.51i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.147 - 0.355i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
19 \( 1 + (0.0317 - 0.0317i)T - 19iT^{2} \)
23 \( 1 + (2.04 + 4.93i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-4.62 - 1.91i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.946 + 2.28i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.90 + 4.59i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.75 - 0.728i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-6.76 - 6.76i)T + 43iT^{2} \)
47 \( 1 + 2.73iT - 47T^{2} \)
53 \( 1 + (-6.87 + 6.87i)T - 53iT^{2} \)
59 \( 1 + (-7.58 - 7.58i)T + 59iT^{2} \)
61 \( 1 + (-12.3 + 5.12i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 4.69T + 67T^{2} \)
71 \( 1 + (3.55 - 8.57i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.02 + 2.08i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.58 + 3.83i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (1.37 - 1.37i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + (-6.88 - 2.85i)T + (68.5 + 68.5i)T^{2} \)
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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

−9.661336017730781098934200475294, −8.783758344797768336761039762162, −8.200549112758727790541170307130, −7.28158748653232634076909275820, −6.45543473717053227349773819118, −5.49849753783661670796755010502, −4.34906003870503064340188799263, −3.65195266884981260970640982326, −2.64296283313921754951680727790, −0.67490892199528815424594477083, 0.67328054559012428692933793527, 2.91186936746385387183974146520, 3.51747951205868498264853709023, 4.23995536016906616789907982626, 5.63184005827182356534136074786, 6.61943587179762728848537365173, 7.32019312466692670224369987061, 7.88388947285219507636795221769, 8.870551574365622599499674640522, 10.02661203129232837548338731603

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