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Properties

Label 2-684-171.103-c2-0-0
Degree $2$
Conductor $684$
Sign $-0.995 + 0.0941i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 2.66i)3-s + (−3.01 − 5.22i)5-s + (−3.98 − 6.90i)7-s + (−5.19 + 7.34i)9-s + (9.13 + 15.8i)11-s + 3.82i·13-s + (9.75 − 15.2i)15-s + (9.74 − 16.8i)17-s + (−10.8 + 15.5i)19-s + (12.9 − 20.1i)21-s − 36.3·23-s + (−5.69 + 9.86i)25-s + (−26.7 − 3.70i)27-s + (−34.0 − 19.6i)29-s + (−13.4 − 7.77i)31-s + ⋯
L(s)  = 1  + (0.459 + 0.888i)3-s + (−0.603 − 1.04i)5-s + (−0.569 − 0.986i)7-s + (−0.577 + 0.816i)9-s + (0.830 + 1.43i)11-s + 0.294i·13-s + (0.650 − 1.01i)15-s + (0.573 − 0.992i)17-s + (−0.572 + 0.819i)19-s + (0.614 − 0.959i)21-s − 1.57·23-s + (−0.227 + 0.394i)25-s + (−0.990 − 0.137i)27-s + (−1.17 − 0.678i)29-s + (−0.434 − 0.250i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.995 + 0.0941i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.995 + 0.0941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1359079878\)
\(L(\frac12)\) \(\approx\) \(0.1359079878\)
\(L(2)\) 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 \( 1 + (-1.37 - 2.66i)T \)
19 \( 1 + (10.8 - 15.5i)T \)
good5 \( 1 + (3.01 + 5.22i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.98 + 6.90i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-9.13 - 15.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 3.82iT - 169T^{2} \)
17 \( 1 + (-9.74 + 16.8i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 36.3T + 529T^{2} \)
29 \( 1 + (34.0 + 19.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (13.4 + 7.77i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 31.2iT - 1.36e3T^{2} \)
41 \( 1 + (-12.6 + 7.31i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 64.4T + 1.84e3T^{2} \)
47 \( 1 + (24.0 - 41.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (88.4 - 51.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (73.8 - 42.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-42.1 + 72.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 31.5iT - 4.48e3T^{2} \)
71 \( 1 + (31.0 + 17.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-33.3 + 57.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 103. iT - 6.24e3T^{2} \)
83 \( 1 + (-57.2 - 99.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (17.9 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 18.4iT - 9.40e3T^{2} \)
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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

−10.44234290286168607771049665954, −9.649337536609078008851707596668, −9.332906841653555238458937355813, −8.063376879008739480660043306187, −7.53132793813746027498080487273, −6.26966913367530483936387005865, −4.83924357464733217428676366165, −4.23892069097794056596680168054, −3.56157920646988067339629714097, −1.76986553195259511026518087657, 0.04338766970130729171693366985, 1.91744476825925827657818885489, 3.21797223561861171414073905033, 3.60682819356116497123186819688, 5.77917912684714539797799476570, 6.28815212384344656701310676645, 7.13462390825412456170714338221, 8.194108446649112744257737241765, 8.711515441188694328288914571199, 9.700435990976742626183446167201

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