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Properties

Label 2-3549-3549.629-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.984 + 0.175i$
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  + (−0.239 + 0.970i)3-s + (0.822 − 0.568i)4-s + (0.935 − 0.354i)7-s + (−0.885 − 0.464i)9-s + (0.354 + 0.935i)12-s − i·13-s + (0.354 − 0.935i)16-s + (0.420 + 0.420i)19-s + (0.120 + 0.992i)21-s + (−0.663 − 0.748i)25-s + (0.663 − 0.748i)27-s + (0.568 − 0.822i)28-s + (0.0217 + 0.359i)31-s + (−0.992 + 0.120i)36-s + (−0.0744 − 1.23i)37-s + ⋯
L(s)  = 1  + (−0.239 + 0.970i)3-s + (0.822 − 0.568i)4-s + (0.935 − 0.354i)7-s + (−0.885 − 0.464i)9-s + (0.354 + 0.935i)12-s − i·13-s + (0.354 − 0.935i)16-s + (0.420 + 0.420i)19-s + (0.120 + 0.992i)21-s + (−0.663 − 0.748i)25-s + (0.663 − 0.748i)27-s + (0.568 − 0.822i)28-s + (0.0217 + 0.359i)31-s + (−0.992 + 0.120i)36-s + (−0.0744 − 1.23i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.984 + 0.175i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.984 + 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.556984572\)
\(L(\frac12)\) \(\approx\) \(1.556984572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.239 - 0.970i)T \)
7 \( 1 + (-0.935 + 0.354i)T \)
13 \( 1 + iT \)
good2 \( 1 + (-0.822 + 0.568i)T^{2} \)
5 \( 1 + (0.663 + 0.748i)T^{2} \)
11 \( 1 + (-0.822 - 0.568i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (-0.420 - 0.420i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.568 - 0.822i)T^{2} \)
31 \( 1 + (-0.0217 - 0.359i)T + (-0.992 + 0.120i)T^{2} \)
37 \( 1 + (0.0744 + 1.23i)T + (-0.992 + 0.120i)T^{2} \)
41 \( 1 + (0.464 - 0.885i)T^{2} \)
43 \( 1 + (-1.31 - 1.48i)T + (-0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.935 - 0.354i)T^{2} \)
53 \( 1 + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (-0.663 - 0.748i)T^{2} \)
61 \( 1 + (1.12 + 0.136i)T + (0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.0217 + 0.118i)T + (-0.935 + 0.354i)T^{2} \)
71 \( 1 + (-0.464 + 0.885i)T^{2} \)
73 \( 1 + (0.244 - 0.783i)T + (-0.822 - 0.568i)T^{2} \)
79 \( 1 + (0.271 - 0.393i)T + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + (-0.464 - 0.885i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.28 + 0.580i)T + (0.663 - 0.748i)T^{2} \)
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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

−8.755527531614691291031912699482, −7.895403720241688628586851145462, −7.38475875033343556408750239230, −6.20935027984536801275455585291, −5.70556300421191244072687777096, −5.00029975114935000002860453761, −4.20270229483108751183754397293, −3.22397594191336771623962321424, −2.28211372776307527250801090737, −1.00407674093642999551874301739, 1.45770086928313226161030435902, 2.11704113864501702191395371070, 2.97305552719105277716382437366, 4.12019343190447037491384914559, 5.14125204775598454068474146261, 5.93952232283083415392558287901, 6.63945539079294221604253918389, 7.42314934555857852779660853047, 7.74280341874197271181931350249, 8.630120417085711172360899591554

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