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Properties

Label 2-1224-17.15-c1-0-22
Degree $2$
Conductor $1224$
Sign $-0.997 + 0.0765i$
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  + (−0.194 + 0.0804i)5-s + (−1.76 − 0.730i)7-s + (−0.356 + 0.861i)11-s − 5.90i·13-s + (−4.07 + 0.626i)17-s + (3.51 + 3.51i)19-s + (−1.71 + 4.14i)23-s + (−3.50 + 3.50i)25-s + (−6.46 + 2.67i)29-s + (−1.31 − 3.17i)31-s + 0.401·35-s + (−1.21 − 2.94i)37-s + (−7.03 − 2.91i)41-s + (−2.78 + 2.78i)43-s − 9.54i·47-s + ⋯
L(s)  = 1  + (−0.0869 + 0.0359i)5-s + (−0.666 − 0.276i)7-s + (−0.107 + 0.259i)11-s − 1.63i·13-s + (−0.988 + 0.151i)17-s + (0.807 + 0.807i)19-s + (−0.357 + 0.864i)23-s + (−0.700 + 0.700i)25-s + (−1.20 + 0.497i)29-s + (−0.235 − 0.569i)31-s + 0.0678·35-s + (−0.200 − 0.484i)37-s + (−1.09 − 0.454i)41-s + (−0.424 + 0.424i)43-s − 1.39i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.997 + 0.0765i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ -0.997 + 0.0765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1440610403\)
\(L(\frac12)\) \(\approx\) \(0.1440610403\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.07 - 0.626i)T \)
good5 \( 1 + (0.194 - 0.0804i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.76 + 0.730i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.356 - 0.861i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.90iT - 13T^{2} \)
19 \( 1 + (-3.51 - 3.51i)T + 19iT^{2} \)
23 \( 1 + (1.71 - 4.14i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (6.46 - 2.67i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.31 + 3.17i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.21 + 2.94i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.03 + 2.91i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.78 - 2.78i)T - 43iT^{2} \)
47 \( 1 + 9.54iT - 47T^{2} \)
53 \( 1 + (2.21 + 2.21i)T + 53iT^{2} \)
59 \( 1 + (1.95 - 1.95i)T - 59iT^{2} \)
61 \( 1 + (0.440 + 0.182i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.33T + 67T^{2} \)
71 \( 1 + (-0.788 - 1.90i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.971 + 0.402i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.85 - 11.7i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.666 + 0.666i)T + 83iT^{2} \)
89 \( 1 + 3.51iT - 89T^{2} \)
97 \( 1 + (2.66 - 1.10i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.535222226919080768547267417827, −8.438080380934797369749124151131, −7.63161136128615561258958744252, −6.98846097320724757836248566861, −5.80822986207345468057301404839, −5.27952271847988957921678069443, −3.82735661817537604550586473918, −3.24485196523131937069218728581, −1.80202373146773350172865268312, −0.05718743257844003992543604137, 1.87644378047934720040101358116, 2.96825939437664641426028244298, 4.12504563291512671957755401355, 4.91097384708059495257024600678, 6.18605430110380418276522144542, 6.67197043006146292225290878652, 7.61017030431864684126486436354, 8.712054211757516953428113995209, 9.249791287085917028317828069840, 9.949068967962198841350963388478

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