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Properties

Label 2-684-171.103-c2-0-23
Degree $2$
Conductor $684$
Sign $0.428 - 0.903i$
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 − 0.176i)3-s + (4.35 + 7.54i)5-s + (5.71 + 9.89i)7-s + (8.93 − 1.05i)9-s + (−8.80 − 15.2i)11-s + 5.93i·13-s + (14.3 + 21.8i)15-s + (10.9 − 18.9i)17-s + (18.1 + 5.77i)19-s + (18.8 + 28.6i)21-s − 2.29·23-s + (−25.4 + 44.0i)25-s + (26.5 − 4.73i)27-s + (26.5 + 15.3i)29-s + (−40.6 − 23.4i)31-s + ⋯
L(s)  = 1  + (0.998 − 0.0586i)3-s + (0.871 + 1.50i)5-s + (0.816 + 1.41i)7-s + (0.993 − 0.117i)9-s + (−0.800 − 1.38i)11-s + 0.456i·13-s + (0.958 + 1.45i)15-s + (0.642 − 1.11i)17-s + (0.952 + 0.303i)19-s + (0.897 + 1.36i)21-s − 0.0997·23-s + (−1.01 + 1.76i)25-s + (0.984 − 0.175i)27-s + (0.915 + 0.528i)29-s + (−1.31 − 0.757i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.428 - 0.903i$
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: \((2,\ 684,\ (\ :1),\ 0.428 - 0.903i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.342770641\)
\(L(\frac12)\) \(\approx\) \(3.342770641\)
\(L(2)\) 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 \( 1 + (-2.99 + 0.176i)T \)
19 \( 1 + (-18.1 - 5.77i)T \)
good5 \( 1 + (-4.35 - 7.54i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.71 - 9.89i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.80 + 15.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.93iT - 169T^{2} \)
17 \( 1 + (-10.9 + 18.9i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 2.29T + 529T^{2} \)
29 \( 1 + (-26.5 - 15.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (40.6 + 23.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 60.1iT - 1.36e3T^{2} \)
41 \( 1 + (28.7 - 16.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 79.2T + 1.84e3T^{2} \)
47 \( 1 + (4.29 - 7.44i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-11.3 + 6.54i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.8 - 13.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (22.2 - 38.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 71.3iT - 4.48e3T^{2} \)
71 \( 1 + (-71.0 - 40.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-0.496 + 0.859i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 73.9iT - 6.24e3T^{2} \)
83 \( 1 + (7.54 + 13.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-118. + 68.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 37.3iT - 9.40e3T^{2} \)
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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

−10.32688695331480180502316015390, −9.500503400906016801564409652831, −8.759453844130821107031331896126, −7.86030028277671900997348224025, −7.04555465103120977244353319828, −5.87741715769130109940781516962, −5.23758323375493895451862322207, −3.30709996406327171754679074769, −2.76041401512526614540489649850, −1.84357470688056906917721805776, 1.19717834383727680107883006091, 1.89554594748563496652138466671, 3.59265783714874194765360720638, 4.81278038597491849008320281825, 5.05518781217032067167868200790, 6.80634524695478671135014111599, 7.931408922225599357504918981826, 8.143447033339190156581259671354, 9.337034273681048873951731488035, 10.10356143990530808108546892822

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