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Properties

Label 2-684-684.535-c1-0-84
Degree $2$
Conductor $684$
Sign $0.440 - 0.897i$
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  + (1.04 + 0.951i)2-s + (1.67 − 0.448i)3-s + (0.189 + 1.99i)4-s + (2.00 + 1.68i)5-s + (2.17 + 1.12i)6-s + (0.592 + 0.342i)7-s + (−1.69 + 2.26i)8-s + (2.59 − 1.50i)9-s + (0.497 + 3.67i)10-s − 6.41i·11-s + (1.20 + 3.24i)12-s + (1.61 + 1.92i)13-s + (0.294 + 0.922i)14-s + (4.11 + 1.91i)15-s + (−3.92 + 0.752i)16-s + (−3.66 − 3.07i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.672i)2-s + (0.965 − 0.259i)3-s + (0.0945 + 0.995i)4-s + (0.897 + 0.753i)5-s + (0.888 + 0.458i)6-s + (0.224 + 0.129i)7-s + (−0.599 + 0.800i)8-s + (0.865 − 0.500i)9-s + (0.157 + 1.16i)10-s − 1.93i·11-s + (0.349 + 0.937i)12-s + (0.447 + 0.533i)13-s + (0.0787 + 0.246i)14-s + (1.06 + 0.494i)15-s + (−0.982 + 0.188i)16-s + (−0.889 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90679 + 1.81223i\)
\(L(\frac12)\) \(\approx\) \(2.90679 + 1.81223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.951i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
19 \( 1 + (4.05 + 1.60i)T \)
good5 \( 1 + (-2.00 - 1.68i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.592 - 0.342i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 6.41iT - 11T^{2} \)
13 \( 1 + (-1.61 - 1.92i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.66 + 3.07i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.09 - 3.00i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.09 - 5.75i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 - 1.22iT - 37T^{2} \)
41 \( 1 + (-9.60 - 1.69i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.178 + 0.489i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.99 - 5.47i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.49 - 0.793i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-9.22 + 3.35i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.709 + 0.595i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.35 - 7.67i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.44 + 8.17i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.01 + 0.732i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-10.0 - 8.43i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-7.31 - 4.22i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.17 - 3.23i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (16.0 + 2.82i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.94538987704328342286366268041, −9.336235474318840602888784103687, −8.835950667118010238541543335930, −7.984482801034898000510422787542, −6.88339878170815582191589931239, −6.33304419360860719548868231496, −5.39239788168767700831567024919, −3.94810100042921579021579194993, −3.07190136056665630623784552463, −2.10700557539917630536681789843, 1.79764784034197102386500494025, 2.21207788569460547234489072269, 3.94509916940418056779269026767, 4.50275540594871205774758160656, 5.51589511175432076437390579767, 6.63856752813224404796104208997, 7.81798469189071477714724532437, 8.968275489215446196756650582329, 9.537330959207739262261228266824, 10.29011804441715672852935933003

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