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Properties

Label 2-3549-3549.2309-c0-0-0
Degree $2$
Conductor $3549$
Sign $-0.505 + 0.862i$
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  + (−0.663 + 0.748i)3-s + (−0.239 − 0.970i)4-s + (−0.354 + 0.935i)7-s + (−0.120 − 0.992i)9-s + (0.885 + 0.464i)12-s − i·13-s + (−0.885 + 0.464i)16-s + (−1.11 + 1.11i)19-s + (−0.464 − 0.885i)21-s + (0.822 − 0.568i)25-s + (0.822 + 0.568i)27-s + (0.992 + 0.120i)28-s + (−0.186 − 1.01i)31-s + (−0.935 + 0.354i)36-s + (−0.328 − 1.79i)37-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)3-s + (−0.239 − 0.970i)4-s + (−0.354 + 0.935i)7-s + (−0.120 − 0.992i)9-s + (0.885 + 0.464i)12-s − i·13-s + (−0.885 + 0.464i)16-s + (−1.11 + 1.11i)19-s + (−0.464 − 0.885i)21-s + (0.822 − 0.568i)25-s + (0.822 + 0.568i)27-s + (0.992 + 0.120i)28-s + (−0.186 − 1.01i)31-s + (−0.935 + 0.354i)36-s + (−0.328 − 1.79i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.505 + 0.862i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3526072758\)
\(L(\frac12)\) \(\approx\) \(0.3526072758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.663 - 0.748i)T \)
7 \( 1 + (0.354 - 0.935i)T \)
13 \( 1 + iT \)
good2 \( 1 + (0.239 + 0.970i)T^{2} \)
5 \( 1 + (-0.822 + 0.568i)T^{2} \)
11 \( 1 + (0.239 - 0.970i)T^{2} \)
17 \( 1 + (0.748 - 0.663i)T^{2} \)
19 \( 1 + (1.11 - 1.11i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (0.186 + 1.01i)T + (-0.935 + 0.354i)T^{2} \)
37 \( 1 + (0.328 + 1.79i)T + (-0.935 + 0.354i)T^{2} \)
41 \( 1 + (-0.992 + 0.120i)T^{2} \)
43 \( 1 + (1.53 - 1.06i)T + (0.354 - 0.935i)T^{2} \)
47 \( 1 + (0.464 - 0.885i)T^{2} \)
53 \( 1 + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.822 - 0.568i)T^{2} \)
61 \( 1 + (1.81 + 0.688i)T + (0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.186 + 0.308i)T + (-0.464 + 0.885i)T^{2} \)
71 \( 1 + (0.992 - 0.120i)T^{2} \)
73 \( 1 + (-1.50 + 1.17i)T + (0.239 - 0.970i)T^{2} \)
79 \( 1 + (1.28 + 0.317i)T + (0.885 + 0.464i)T^{2} \)
83 \( 1 + (0.992 + 0.120i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.420 + 1.35i)T + (-0.822 - 0.568i)T^{2} \)
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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

−8.736560495124788063640570924091, −7.917738201356140621037538840537, −6.56830531815614931012755245686, −6.09099171128896275128526393811, −5.53862520723953756073302974727, −4.85650620340801561126432129041, −4.00875646923177154884864342191, −2.98957984543207697580648441812, −1.80081269825433796701421780497, −0.22761969172063096905199092358, 1.39185398060350387900262518158, 2.59889956342373722868572657775, 3.54511667315644606819864481249, 4.53575953211962011194507238619, 4.99206207120132870843546774813, 6.38047622129844106902844276954, 6.95659067717162838840141385985, 7.16676615834751214786050953098, 8.290730934031100953249308980490, 8.716801145165271818971599612898

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