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Properties

Label 2-3549-3549.2411-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.431 + 0.902i$
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  + (−0.160 − 0.987i)3-s + (0.992 − 0.120i)4-s + (−0.960 + 0.278i)7-s + (−0.948 + 0.316i)9-s + (−0.278 − 0.960i)12-s + (0.866 + 0.5i)13-s + (0.970 − 0.239i)16-s + (1.61 − 0.431i)19-s + (0.428 + 0.903i)21-s + (−0.534 − 0.845i)25-s + (0.464 + 0.885i)27-s + (−0.919 + 0.391i)28-s + (0.906 − 0.836i)31-s + (−0.903 + 0.428i)36-s + (0.641 + 0.199i)37-s + ⋯
L(s)  = 1  + (−0.160 − 0.987i)3-s + (0.992 − 0.120i)4-s + (−0.960 + 0.278i)7-s + (−0.948 + 0.316i)9-s + (−0.278 − 0.960i)12-s + (0.866 + 0.5i)13-s + (0.970 − 0.239i)16-s + (1.61 − 0.431i)19-s + (0.428 + 0.903i)21-s + (−0.534 − 0.845i)25-s + (0.464 + 0.885i)27-s + (−0.919 + 0.391i)28-s + (0.906 − 0.836i)31-s + (−0.903 + 0.428i)36-s + (0.641 + 0.199i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.431 + 0.902i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441134316\)
\(L(\frac12)\) \(\approx\) \(1.441134316\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.160 + 0.987i)T \)
7 \( 1 + (0.960 - 0.278i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.992 + 0.120i)T^{2} \)
5 \( 1 + (0.534 + 0.845i)T^{2} \)
11 \( 1 + (0.600 - 0.799i)T^{2} \)
17 \( 1 + (0.354 + 0.935i)T^{2} \)
19 \( 1 + (-1.61 + 0.431i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.799 + 0.600i)T^{2} \)
31 \( 1 + (-0.906 + 0.836i)T + (0.0804 - 0.996i)T^{2} \)
37 \( 1 + (-0.641 - 0.199i)T + (0.822 + 0.568i)T^{2} \)
41 \( 1 + (0.979 + 0.200i)T^{2} \)
43 \( 1 + (1.80 - 0.0727i)T + (0.996 - 0.0804i)T^{2} \)
47 \( 1 + (0.721 + 0.692i)T^{2} \)
53 \( 1 + (0.632 - 0.774i)T^{2} \)
59 \( 1 + (-0.464 + 0.885i)T^{2} \)
61 \( 1 + (0.148 + 1.83i)T + (-0.987 + 0.160i)T^{2} \)
67 \( 1 + (-0.712 + 1.77i)T + (-0.721 - 0.692i)T^{2} \)
71 \( 1 + (0.316 + 0.948i)T^{2} \)
73 \( 1 + (-1.66 - 1.10i)T + (0.391 + 0.919i)T^{2} \)
79 \( 1 + (1.72 - 0.732i)T + (0.692 - 0.721i)T^{2} \)
83 \( 1 + (0.663 + 0.748i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.0389 - 1.93i)T + (-0.999 - 0.0402i)T^{2} \)
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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

−8.317875038536719105609207159881, −7.84755747587678021186962677492, −6.87143940453123428988836588370, −6.51459001756902743613396749778, −5.92957748069097363929176157640, −5.11376861124758635464468754899, −3.62696708218835315271663089141, −2.88539531476636066013589193938, −2.07668286910820845905856872152, −0.973739223477951902221006321456, 1.25976649243459428705421408178, 2.87440105991608598528494226712, 3.30438808588292039926033479077, 4.02486516464968917393981534490, 5.30897573978721526019672636842, 5.83185131191278427273040180128, 6.56493441567950735499974915833, 7.33449083842630925514921539107, 8.152859574127864870727695841482, 8.947773276727493931072773386061

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