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Properties

Label 2-684-171.103-c2-0-15
Degree $2$
Conductor $684$
Sign $0.348 + 0.937i$
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  + (−2.99 + 0.0790i)3-s + (−2.76 − 4.78i)5-s + (0.838 + 1.45i)7-s + (8.98 − 0.473i)9-s + (0.764 + 1.32i)11-s + 20.7i·13-s + (8.65 + 14.1i)15-s + (9.66 − 16.7i)17-s + (−16.1 + 10.0i)19-s + (−2.62 − 4.28i)21-s + 14.7·23-s + (−2.73 + 4.74i)25-s + (−26.9 + 2.13i)27-s + (35.3 + 20.4i)29-s + (−23.1 − 13.3i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0263i)3-s + (−0.552 − 0.956i)5-s + (0.119 + 0.207i)7-s + (0.998 − 0.0526i)9-s + (0.0694 + 0.120i)11-s + 1.59i·13-s + (0.577 + 0.941i)15-s + (0.568 − 0.985i)17-s + (−0.850 + 0.526i)19-s + (−0.125 − 0.204i)21-s + 0.639·23-s + (−0.109 + 0.189i)25-s + (−0.996 + 0.0789i)27-s + (1.21 + 0.703i)29-s + (−0.745 − 0.430i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.348 + 0.937i$
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: \((2,\ 684,\ (\ :1),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9655731405\)
\(L(\frac12)\) \(\approx\) \(0.9655731405\)
\(L(2)\) 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 \( 1 + (2.99 - 0.0790i)T \)
19 \( 1 + (16.1 - 10.0i)T \)
good5 \( 1 + (2.76 + 4.78i)T + (-12.5 + 21.6i)T^{2} \)
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 \( 1 - 20.7iT - 169T^{2} \)
17 \( 1 + (-9.66 + 16.7i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 14.7T + 529T^{2} \)
29 \( 1 + (-35.3 - 20.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.1 + 13.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 59.7iT - 1.36e3T^{2} \)
41 \( 1 + (31.9 - 18.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 4.76T + 1.84e3T^{2} \)
47 \( 1 + (-27.0 + 46.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (17.5 - 10.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-85.6 + 49.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-43.6 + 75.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 86.3iT - 4.48e3T^{2} \)
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 \( 1 + 36.0iT - 6.24e3T^{2} \)
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 \( 1 + 48.3iT - 9.40e3T^{2} \)
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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

−10.15477354342406435664644484025, −9.210860158489787454318479668154, −8.515080820286350456623967846367, −7.29960882174506521170416797280, −6.60112787959880752496690330282, −5.39771110690971940823772999582, −4.69334843484761626644230937502, −3.87126516857941855059250056992, −1.86618357294988438284695592703, −0.52113963433203613606373390416, 0.981993474905204806046826843327, 2.86366352742290716534623827928, 3.93200358150546749899540201217, 5.07742638650639655784128529905, 6.04974059806095044202568947091, 6.85108954449267737081640990950, 7.67332396448921681963742465350, 8.529082181484511479200026685624, 10.08878647297942735324684709331, 10.53875687010473117794725244236

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