Properties

Label 2-3549-3549.236-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.735 - 0.677i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.721 − 0.692i)3-s + (0.822 − 0.568i)4-s + (−0.160 + 0.987i)7-s + (0.0402 + 0.999i)9-s + (−0.987 − 0.160i)12-s + (−0.866 + 0.5i)13-s + (0.354 − 0.935i)16-s + (−0.504 + 1.88i)19-s + (0.799 − 0.600i)21-s + (−0.316 + 0.948i)25-s + (0.663 − 0.748i)27-s + (0.428 + 0.903i)28-s + (0.300 − 0.198i)31-s + (0.600 + 0.799i)36-s + (0.119 + 1.97i)37-s + ⋯
L(s)  = 1  + (−0.721 − 0.692i)3-s + (0.822 − 0.568i)4-s + (−0.160 + 0.987i)7-s + (0.0402 + 0.999i)9-s + (−0.987 − 0.160i)12-s + (−0.866 + 0.5i)13-s + (0.354 − 0.935i)16-s + (−0.504 + 1.88i)19-s + (0.799 − 0.600i)21-s + (−0.316 + 0.948i)25-s + (0.663 − 0.748i)27-s + (0.428 + 0.903i)28-s + (0.300 − 0.198i)31-s + (0.600 + 0.799i)36-s + (0.119 + 1.97i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9578319647\)
\(L(\frac12)\) \(\approx\) \(0.9578319647\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.721 + 0.692i)T \)
7 \( 1 + (0.160 - 0.987i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.822 + 0.568i)T^{2} \)
5 \( 1 + (0.316 - 0.948i)T^{2} \)
11 \( 1 + (-0.0804 + 0.996i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (0.504 - 1.88i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.996 - 0.0804i)T^{2} \)
31 \( 1 + (-0.300 + 0.198i)T + (0.391 - 0.919i)T^{2} \)
37 \( 1 + (-0.119 - 1.97i)T + (-0.992 + 0.120i)T^{2} \)
41 \( 1 + (0.534 + 0.845i)T^{2} \)
43 \( 1 + (-1.17 + 0.240i)T + (0.919 - 0.391i)T^{2} \)
47 \( 1 + (-0.774 - 0.632i)T^{2} \)
53 \( 1 + (-0.278 + 0.960i)T^{2} \)
59 \( 1 + (-0.663 - 0.748i)T^{2} \)
61 \( 1 + (-0.336 - 0.788i)T + (-0.692 + 0.721i)T^{2} \)
67 \( 1 + (-0.113 - 0.0405i)T + (0.774 + 0.632i)T^{2} \)
71 \( 1 + (0.999 - 0.0402i)T^{2} \)
73 \( 1 + (1.94 - 0.437i)T + (0.903 - 0.428i)T^{2} \)
79 \( 1 + (0.205 + 0.432i)T + (-0.632 + 0.774i)T^{2} \)
83 \( 1 + (-0.464 - 0.885i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.0520 - 0.515i)T + (-0.979 - 0.200i)T^{2} \)
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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

−8.735803051926116444967131653840, −7.88571273985254066975038290561, −7.25501167795639871196276012774, −6.46862183802766314343872829872, −5.89373000830750681446017969552, −5.41279217715684958874303532095, −4.45547205833839170596051115958, −3.01668120757810383059749222778, −2.11930224232846306341991006743, −1.45421331970783444125417464320, 0.59056737380499673850550789737, 2.32568037990883401648469757462, 3.17317025841011757224997506609, 4.14859953947148832532825946297, 4.66995323220170446381288331871, 5.72019615855751958062124905384, 6.51410627639396652381254345973, 7.13249669613202932115441578696, 7.67076240970671727452550530542, 8.715924838104211522158701464797

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