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Properties

Label 2-1224-17.4-c1-0-7
Degree $2$
Conductor $1224$
Sign $0.615 - 0.788i$
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  + (1 − i)5-s + (2 + 2i)7-s + (2 + 2i)11-s − 4·13-s + (1 + 4i)17-s + 8i·19-s + (−2 − 2i)23-s + 3i·25-s + (3 − 3i)29-s + (6 − 6i)31-s + 4·35-s + (−5 + 5i)37-s + (−1 − i)41-s − 8i·43-s + i·49-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + (0.755 + 0.755i)7-s + (0.603 + 0.603i)11-s − 1.10·13-s + (0.242 + 0.970i)17-s + 1.83i·19-s + (−0.417 − 0.417i)23-s + 0.600i·25-s + (0.557 − 0.557i)29-s + (1.07 − 1.07i)31-s + 0.676·35-s + (−0.821 + 0.821i)37-s + (−0.156 − 0.156i)41-s − 1.21i·43-s + 0.142i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.833902490\)
\(L(\frac12)\) \(\approx\) \(1.833902490\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 4i)T \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + (-2 - 2i)T + 11iT^{2} \)
13 \( 1 + 4T + 13T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + (2 + 2i)T + 23iT^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + (-6 + 6i)T - 31iT^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 + (1 + i)T + 41iT^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (-6 + 6i)T - 71iT^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 + (2 + 2i)T + 79iT^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−9.950992718149234160333523946038, −8.992068170357823798352967295331, −8.241140964231758419917645870570, −7.56748276825542485835839369170, −6.34115583552783806891136225798, −5.62588177234321767236045425242, −4.77961239872255020688562671716, −3.86375521000672725990504983931, −2.31080133021042445187302819978, −1.53249864358340460390757576502, 0.825902666573810959315898493091, 2.31573472974289544915128209932, 3.29573052222550213543115332064, 4.64808978263069280436016960670, 5.13474322475783092014204763462, 6.54424462863782576758474608585, 7.00411520041302951401685462022, 7.912073971212656658293767892601, 8.840472931895572960922466664843, 9.692105070978684921751132248115

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