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Properties

Label 2-684-171.31-c2-0-4
Degree $2$
Conductor $684$
Sign $-0.910 - 0.412i$
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  + (−1.43 + 2.63i)3-s + 5.52·5-s + (0.838 + 1.45i)7-s + (−4.90 − 7.54i)9-s + (0.764 + 1.32i)11-s + (−17.9 + 10.3i)13-s + (−7.90 + 14.5i)15-s + (9.66 + 16.7i)17-s + (−16.1 − 10.0i)19-s + (−5.02 + 0.132i)21-s + (−7.35 − 12.7i)23-s + 5.47·25-s + (26.9 − 2.13i)27-s + 40.8i·29-s + (23.1 + 13.3i)31-s + ⋯
L(s)  = 1  + (−0.477 + 0.878i)3-s + 1.10·5-s + (0.119 + 0.207i)7-s + (−0.544 − 0.838i)9-s + (0.0694 + 0.120i)11-s + (−1.38 + 0.797i)13-s + (−0.526 + 0.970i)15-s + (0.568 + 0.985i)17-s + (−0.850 − 0.526i)19-s + (−0.239 + 0.00630i)21-s + (−0.319 − 0.553i)23-s + 0.219·25-s + (0.996 − 0.0789i)27-s + 1.40i·29-s + (0.745 + 0.430i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.910 - 0.412i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.910 - 0.412i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.098428643\)
\(L(\frac12)\) \(\approx\) \(1.098428643\)
\(L(2)\) 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 \( 1 + (1.43 - 2.63i)T \)
19 \( 1 + (16.1 + 10.0i)T \)
good5 \( 1 - 5.52T + 25T^{2} \)
7 \( 1 + (-0.838 - 1.45i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.764 - 1.32i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (17.9 - 10.3i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-9.66 - 16.7i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (7.35 + 12.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 40.8iT - 841T^{2} \)
31 \( 1 + (-23.1 - 13.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 59.7iT - 1.36e3T^{2} \)
41 \( 1 - 36.9iT - 1.68e3T^{2} \)
43 \( 1 + (-2.38 + 4.12i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 54.0T + 2.20e3T^{2} \)
53 \( 1 + (17.5 + 10.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 98.9iT - 3.48e3T^{2} \)
61 \( 1 + 87.2T + 3.72e3T^{2} \)
67 \( 1 + (74.7 - 43.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (76.0 - 43.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-30.5 - 52.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (31.1 + 18.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.38 - 7.58i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-63.7 - 36.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (41.8 + 24.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.30740500013701487599108019799, −9.996514221760148925416669920119, −9.148910814099759068536170204931, −8.314463392753863048619329952397, −6.77939132471246650408684668600, −6.16915961912697106226749772760, −5.09037330950395851954353189834, −4.48831610774641935093999275518, −3.01091655780935519385355584260, −1.74220696642227698406313314717, 0.38961820150163035994092402230, 1.87198711481880901466333336444, 2.76499133562258852376755951556, 4.60434382645674170170880057168, 5.67383715453885748138009512774, 6.10708303168000296017733956588, 7.40826855202807383598340193164, 7.80671732015559132462814679040, 9.167253089833697475201129539493, 10.01203983218560177179827166630

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