Properties

Label 2-4368-13.12-c1-0-30
Degree $2$
Conductor $4368$
Sign $0.554 - 0.832i$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  − 3-s + i·5-s i·7-s + 9-s + 3i·11-s + (3 + 2i)13-s i·15-s + 7·17-s + 3i·19-s + i·21-s − 23-s + 4·25-s − 27-s − 29-s − 8i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447i·5-s − 0.377i·7-s + 0.333·9-s + 0.904i·11-s + (0.832 + 0.554i)13-s − 0.258i·15-s + 1.69·17-s + 0.688i·19-s + 0.218i·21-s − 0.208·23-s + 0.800·25-s − 0.192·27-s − 0.185·29-s − 1.43i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4368} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690887109\)
\(L(\frac12)\) \(\approx\) \(1.690887109\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + iT \)
13 \( 1 + (-3 - 2i)T \)
good5 \( 1 - iT - 5T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 13iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.349767129780511638535051620321, −7.54664289122515777609877662114, −7.13675936219224482208945724592, −6.15677583157293307219287170823, −5.74480894316983856240713556968, −4.69560488741758750890461214685, −3.98620155066313285100279508886, −3.19281409100618000644348155891, −1.94683708157462223591076479602, −0.995126112174070782269970225459, 0.66854931075850297804195872741, 1.46423731735087239600263554104, 2.99781194192239128282687873459, 3.52127693754453556156401590873, 4.75338849251651190930964375180, 5.33017364760809383556233696477, 5.98520933629242487396428346402, 6.59892478696932866614096836323, 7.67485510987562422449112944898, 8.209662568024948702209953936941

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