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Properties

Label 2-684-171.103-c2-0-6
Degree $2$
Conductor $684$
Sign $-0.992 - 0.121i$
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  + (−0.0757 + 2.99i)3-s + (4.34 + 7.52i)5-s + (0.405 + 0.702i)7-s + (−8.98 − 0.454i)9-s + (−1.31 − 2.27i)11-s + 4.28i·13-s + (−22.9 + 12.4i)15-s + (−5.20 + 9.00i)17-s + (3.39 + 18.6i)19-s + (−2.13 + 1.16i)21-s + 9.94·23-s + (−25.2 + 43.8i)25-s + (2.04 − 26.9i)27-s + (−27.9 − 16.1i)29-s + (15.6 + 9.04i)31-s + ⋯
L(s)  = 1  + (−0.0252 + 0.999i)3-s + (0.869 + 1.50i)5-s + (0.0579 + 0.100i)7-s + (−0.998 − 0.0505i)9-s + (−0.119 − 0.207i)11-s + 0.329i·13-s + (−1.52 + 0.831i)15-s + (−0.305 + 0.529i)17-s + (0.178 + 0.983i)19-s + (−0.101 + 0.0553i)21-s + 0.432·23-s + (−1.01 + 1.75i)25-s + (0.0757 − 0.997i)27-s + (−0.962 − 0.555i)29-s + (0.505 + 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.992 - 0.121i$
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: \((2,\ 684,\ (\ :1),\ -0.992 - 0.121i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.658859087\)
\(L(\frac12)\) \(\approx\) \(1.658859087\)
\(L(2)\) 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 \( 1 + (0.0757 - 2.99i)T \)
19 \( 1 + (-3.39 - 18.6i)T \)
good5 \( 1 + (-4.34 - 7.52i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.405 - 0.702i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.31 + 2.27i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.28iT - 169T^{2} \)
17 \( 1 + (5.20 - 9.00i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 9.94T + 529T^{2} \)
29 \( 1 + (27.9 + 16.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-15.6 - 9.04i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 7.21iT - 1.36e3T^{2} \)
41 \( 1 + (-32.4 + 18.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 22.7T + 1.84e3T^{2} \)
47 \( 1 + (-31.0 + 53.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (69.3 - 40.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-16.2 + 9.38i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (13.7 - 23.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 42.9iT - 4.48e3T^{2} \)
71 \( 1 + (29.6 + 17.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-28.0 + 48.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 + (-1.57 - 2.72i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (51.0 - 29.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 60.0iT - 9.40e3T^{2} \)
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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

−10.61840904707533318422552082852, −9.959012586344544355827637309942, −9.259289705292481399977189715805, −8.195526950089453098996072931887, −7.01847628540379325598132182722, −6.09433801902757093607268945696, −5.46094354076508898287018892530, −4.04720978262519373053439478583, −3.13196920048728676121973634550, −2.09759903077158504515325678743, 0.58108318894941695005598364003, 1.61030524047408929532120589529, 2.76613049532409709224239875643, 4.60705167023692481560387630743, 5.35748814913619930704635704151, 6.20904886454567091572997746771, 7.28070646326038153530955352596, 8.144633062665785032319912343452, 9.068975549720693252637238913030, 9.494346821077246830300492058923

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