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Properties

Label 2-1224-17.15-c1-0-1
Degree $2$
Conductor $1224$
Sign $-0.586 - 0.810i$
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  + (0.429 − 0.177i)5-s + (−1.62 − 0.671i)7-s + (−1.79 + 4.33i)11-s − 0.542i·13-s + (2.66 + 3.14i)17-s + (−5.16 − 5.16i)19-s + (−1.25 + 3.03i)23-s + (−3.38 + 3.38i)25-s + (−1.78 + 0.740i)29-s + (0.712 + 1.71i)31-s − 0.815·35-s + (1.30 + 3.16i)37-s + (−2.06 − 0.854i)41-s + (−2.49 + 2.49i)43-s + 10.6i·47-s + ⋯
L(s)  = 1  + (0.192 − 0.0795i)5-s + (−0.612 − 0.253i)7-s + (−0.541 + 1.30i)11-s − 0.150i·13-s + (0.646 + 0.763i)17-s + (−1.18 − 1.18i)19-s + (−0.261 + 0.632i)23-s + (−0.676 + 0.676i)25-s + (−0.331 + 0.137i)29-s + (0.127 + 0.308i)31-s − 0.137·35-s + (0.215 + 0.519i)37-s + (−0.322 − 0.133i)41-s + (−0.380 + 0.380i)43-s + 1.55i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.586 - 0.810i$
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: \((2,\ 1224,\ (\ :1/2),\ -0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7303701495\)
\(L(\frac12)\) \(\approx\) \(0.7303701495\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.66 - 3.14i)T \)
good5 \( 1 + (-0.429 + 0.177i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.62 + 0.671i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.79 - 4.33i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.542iT - 13T^{2} \)
19 \( 1 + (5.16 + 5.16i)T + 19iT^{2} \)
23 \( 1 + (1.25 - 3.03i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.78 - 0.740i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.712 - 1.71i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.30 - 3.16i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.06 + 0.854i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.49 - 2.49i)T - 43iT^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 + (0.634 + 0.634i)T + 53iT^{2} \)
59 \( 1 + (7.78 - 7.78i)T - 59iT^{2} \)
61 \( 1 + (-13.1 - 5.46i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 5.80T + 67T^{2} \)
71 \( 1 + (2.87 + 6.93i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.69 - 1.11i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.62 - 6.34i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (7.82 + 7.82i)T + 83iT^{2} \)
89 \( 1 - 1.76iT - 89T^{2} \)
97 \( 1 + (-0.944 + 0.391i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.946695352508077507931678391767, −9.389321901238536315852596887891, −8.348995403391411149002759494983, −7.49043635273956056888315361327, −6.76500548394779932830746508292, −5.84802993947672454217524106706, −4.87316366538164713534748014432, −3.96897709026415352783582547793, −2.81317802999086942661496456404, −1.64862031927124300177314143102, 0.29334651708571257071770460862, 2.12358988679464247621796434964, 3.17370854083185211883810737504, 4.08618516147699596862881068783, 5.42798923346110158153131900290, 6.02802633777685760087493500953, 6.81005061119282871409920403328, 8.060305758680979032881262634463, 8.454339436039799345119601006741, 9.570237744007696943269978106373

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