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Properties

Label 2-684-171.121-c1-0-0
Degree $2$
Conductor $684$
Sign $-0.626 + 0.779i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
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  + (0.438 + 1.67i)3-s − 4.24·5-s + (1.38 + 2.39i)7-s + (−2.61 + 1.46i)9-s + (−1.31 − 2.26i)11-s + (0.0473 + 0.0819i)13-s + (−1.86 − 7.11i)15-s + (−0.170 − 0.295i)17-s + (−1.77 − 3.97i)19-s + (−3.40 + 3.36i)21-s + (−2.31 − 4.00i)23-s + 13.0·25-s + (−3.60 − 3.73i)27-s − 1.41·29-s + (1.43 − 2.48i)31-s + ⋯
L(s)  = 1  + (0.253 + 0.967i)3-s − 1.89·5-s + (0.522 + 0.904i)7-s + (−0.871 + 0.489i)9-s + (−0.395 − 0.684i)11-s + (0.0131 + 0.0227i)13-s + (−0.480 − 1.83i)15-s + (−0.0413 − 0.0715i)17-s + (−0.408 − 0.912i)19-s + (−0.743 + 0.734i)21-s + (−0.481 − 0.834i)23-s + 2.60·25-s + (−0.694 − 0.719i)27-s − 0.262·29-s + (0.257 − 0.446i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.626 + 0.779i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.626 + 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0316906 - 0.0661454i\)
\(L(\frac12)\) \(\approx\) \(0.0316906 - 0.0661454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.438 - 1.67i)T \)
19 \( 1 + (1.77 + 3.97i)T \)
good5 \( 1 + 4.24T + 5T^{2} \)
7 \( 1 + (-1.38 - 2.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.31 + 2.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0473 - 0.0819i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.170 + 0.295i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.31 + 4.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 + (-1.43 + 2.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + 2.42T + 41T^{2} \)
43 \( 1 + (6.27 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.84T + 47T^{2} \)
53 \( 1 + (2.21 - 3.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 1.31T + 61T^{2} \)
67 \( 1 + (3.61 + 6.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.00 - 8.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.76 + 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.76 - 3.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.93 + 3.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.91 - 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.215 + 0.372i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19692440085656052887645515326, −10.34066811562464958704387195277, −9.051110186939982496408740925894, −8.353111608540927401215454404655, −8.027051473097345366965731519021, −6.69236786856316715151489007195, −5.27291261176616063296142169729, −4.56916384930894147108335612076, −3.61049419283752923517228732757, −2.67428135643817164663194278435, 0.03734148335299189123193408063, 1.64727284345173477331605310007, 3.37983789352787473913997284555, 4.09085877496720763884091361291, 5.27271590766200390022605600254, 6.95044814383559917992024059068, 7.25018781989764722318063149441, 8.166923238312710401075195578467, 8.488361166787791782356373595456, 10.10007765452753571560889812361

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