Properties

Label 2-1224-136.59-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.651 - 0.758i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.292i)22-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 − 0.292i)41-s + (1 + i)43-s + (0.707 − 0.292i)44-s + (0.707 − 0.707i)49-s − 1.00·50-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.292i)22-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 − 0.292i)41-s + (1 + i)43-s + (0.707 − 0.292i)44-s + (0.707 − 0.707i)49-s − 1.00·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.651 - 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7764748568\)
\(L(\frac12)\) \(\approx\) \(0.7764748568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
17 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
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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

−9.791875644756879987676109144742, −9.241048436707080479942676643124, −8.416967790788477120431330643484, −7.42918316049116237961349697330, −7.02395623210018768690719480898, −5.93821920624968620100425069385, −5.16005311581124716762166076523, −4.18594657185239371811408309849, −2.65711377673224757773546319676, −1.26150776971721514249157610485, 1.09538257302704449829476336080, 2.46738072629543346043103425127, 3.48628519714549319517541982772, 4.34860258967616613184573277227, 5.66618825104327680111042148325, 6.64141875690812821166773048054, 7.60762134912791421563127979829, 8.366050349721793530214927732660, 9.011715341000222094396673510382, 9.848518975973238973584302436923

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