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Properties

Label 2-684-4.3-c2-0-61
Degree $2$
Conductor $684$
Sign $0.588 - 0.808i$
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  + (1.90 + 0.618i)2-s + (3.23 + 2.35i)4-s + 8.79·5-s + 0.210i·7-s + (4.69 + 6.47i)8-s + (16.7 + 5.44i)10-s + 12.4i·11-s + 7.39·13-s + (−0.130 + 0.400i)14-s + (4.91 + 15.2i)16-s − 28.0·17-s + 4.35i·19-s + (28.4 + 20.7i)20-s + (−7.69 + 23.6i)22-s − 22.0i·23-s + ⋯
L(s)  = 1  + (0.950 + 0.309i)2-s + (0.808 + 0.588i)4-s + 1.75·5-s + 0.0300i·7-s + (0.586 + 0.809i)8-s + (1.67 + 0.544i)10-s + 1.13i·11-s + 0.569·13-s + (−0.00931 + 0.0286i)14-s + (0.307 + 0.951i)16-s − 1.65·17-s + 0.229i·19-s + (1.42 + 1.03i)20-s + (−0.349 + 1.07i)22-s − 0.960i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.588 - 0.808i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.588 - 0.808i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.521488500\)
\(L(\frac12)\) \(\approx\) \(4.521488500\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.90 - 0.618i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 - 8.79T + 25T^{2} \)
7 \( 1 - 0.210iT - 49T^{2} \)
11 \( 1 - 12.4iT - 121T^{2} \)
13 \( 1 - 7.39T + 169T^{2} \)
17 \( 1 + 28.0T + 289T^{2} \)
23 \( 1 + 22.0iT - 529T^{2} \)
29 \( 1 + 23.1T + 841T^{2} \)
31 \( 1 + 46.6iT - 961T^{2} \)
37 \( 1 - 5.46T + 1.36e3T^{2} \)
41 \( 1 - 10.6T + 1.68e3T^{2} \)
43 \( 1 + 65.4iT - 1.84e3T^{2} \)
47 \( 1 - 75.5iT - 2.20e3T^{2} \)
53 \( 1 + 6.19T + 2.80e3T^{2} \)
59 \( 1 + 50.0iT - 3.48e3T^{2} \)
61 \( 1 + 75.1T + 3.72e3T^{2} \)
67 \( 1 + 63.2iT - 4.48e3T^{2} \)
71 \( 1 + 27.7iT - 5.04e3T^{2} \)
73 \( 1 - 21.8T + 5.32e3T^{2} \)
79 \( 1 - 108. iT - 6.24e3T^{2} \)
83 \( 1 - 26.3iT - 6.88e3T^{2} \)
89 \( 1 + 15.1T + 7.92e3T^{2} \)
97 \( 1 - 117.T + 9.40e3T^{2} \)
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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

−10.53278774366501224395598812971, −9.540564582742346051014208244424, −8.782419542330785086993677140823, −7.48014637084312054279610244834, −6.49189476715322269032705975897, −6.02202034581851727725529026132, −4.99164256474637857744792850881, −4.12920672484533140702610923293, −2.47143369433541239100183443614, −1.89502874957819573453504144079, 1.32103478980296015986065721541, 2.35406819485517991271545922321, 3.41290958442809913335184343675, 4.76495058489010584248996519922, 5.71192335228350983356354032134, 6.21597013764631916542145384422, 7.08449837948978056460975776416, 8.719523586834669664529833923082, 9.362954871114177452828148118749, 10.43257361710610038915039532473

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