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Properties

Label 2-1224-8.5-c1-0-4
Degree $2$
Conductor $1224$
Sign $-0.384 + 0.923i$
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  + (−0.185 + 1.40i)2-s + (−1.93 − 0.520i)4-s + 3.84i·5-s − 1.15·7-s + (1.08 − 2.61i)8-s + (−5.38 − 0.713i)10-s + 2.52i·11-s − 0.601i·13-s + (0.214 − 1.62i)14-s + (3.45 + 2.00i)16-s − 17-s + 7.42i·19-s + (2.00 − 7.42i)20-s + (−3.54 − 0.469i)22-s − 5.19·23-s + ⋯
L(s)  = 1  + (−0.131 + 0.991i)2-s + (−0.965 − 0.260i)4-s + 1.71i·5-s − 0.437·7-s + (0.384 − 0.923i)8-s + (−1.70 − 0.225i)10-s + 0.762i·11-s − 0.166i·13-s + (0.0573 − 0.433i)14-s + (0.864 + 0.502i)16-s − 0.242·17-s + 1.70i·19-s + (0.447 − 1.66i)20-s + (−0.756 − 0.100i)22-s − 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.384 + 0.923i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ -0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6438245541\)
\(L(\frac12)\) \(\approx\) \(0.6438245541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.185 - 1.40i)T \)
3 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 3.84iT - 5T^{2} \)
7 \( 1 + 1.15T + 7T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 + 0.601iT - 13T^{2} \)
19 \( 1 - 7.42iT - 19T^{2} \)
23 \( 1 + 5.19T + 23T^{2} \)
29 \( 1 + 3.84iT - 29T^{2} \)
31 \( 1 - 3.89T + 31T^{2} \)
37 \( 1 + 5.94iT - 37T^{2} \)
41 \( 1 + 8.77T + 41T^{2} \)
43 \( 1 + 2.75iT - 43T^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 + 6.26iT - 53T^{2} \)
59 \( 1 - 6.81iT - 59T^{2} \)
61 \( 1 - 3.84iT - 61T^{2} \)
67 \( 1 + 1.16iT - 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 7.09T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 9.76iT - 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 7.29T + 97T^{2} \)
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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

−10.05671621081805689805132930623, −9.712091247739161210488039391922, −8.376818764788126692713381750798, −7.64191080943679620668463522863, −6.97190274643569759598904655954, −6.26765913294058349349971654645, −5.64014892702060883206856140737, −4.18628679851879349965342679962, −3.46013086426460535301612491284, −2.10946896161342768600355726021, 0.29884304102875100521994951937, 1.38314724309166161412235040496, 2.72473966592247212393135909163, 3.89670850696967669888883161633, 4.75485878419819140377536005863, 5.38515136666584675563535284028, 6.59165750940984016923741901608, 8.043963109803714483879370166889, 8.556836063366813248869339643970, 9.242862327138719545017039870227

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