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Properties

Label 2-684-12.11-c1-0-23
Degree $2$
Conductor $684$
Sign $-0.275 + 0.961i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 0.818i)2-s + (0.660 + 1.88i)4-s − 3.65i·5-s − 0.320i·7-s + (0.783 − 2.71i)8-s + (−2.99 + 4.21i)10-s + 6.11·11-s + 4.17·13-s + (−0.262 + 0.369i)14-s + (−3.12 + 2.49i)16-s + 2.63i·17-s − i·19-s + (6.90 − 2.41i)20-s + (−7.05 − 5.00i)22-s − 4.43·23-s + ⋯
L(s)  = 1  + (−0.815 − 0.578i)2-s + (0.330 + 0.943i)4-s − 1.63i·5-s − 0.121i·7-s + (0.277 − 0.960i)8-s + (−0.947 + 1.33i)10-s + 1.84·11-s + 1.15·13-s + (−0.0700 + 0.0987i)14-s + (−0.782 + 0.623i)16-s + 0.640i·17-s − 0.229i·19-s + (1.54 − 0.540i)20-s + (−1.50 − 1.06i)22-s − 0.924·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.275 + 0.961i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679863 - 0.902050i\)
\(L(\frac12)\) \(\approx\) \(0.679863 - 0.902050i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.818i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 + 3.65iT - 5T^{2} \)
7 \( 1 + 0.320iT - 7T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
13 \( 1 - 4.17T + 13T^{2} \)
17 \( 1 - 2.63iT - 17T^{2} \)
23 \( 1 + 4.43T + 23T^{2} \)
29 \( 1 + 1.67iT - 29T^{2} \)
31 \( 1 + 1.39iT - 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 + 7.88iT - 41T^{2} \)
43 \( 1 - 0.224iT - 43T^{2} \)
47 \( 1 + 9.44T + 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 - 7.49iT - 67T^{2} \)
71 \( 1 - 3.81T + 71T^{2} \)
73 \( 1 + 4.29T + 73T^{2} \)
79 \( 1 - 3.82iT - 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 - 1.06T + 97T^{2} \)
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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

−9.976865537711708895708076281718, −9.286373232133644223548095960424, −8.612045132675366591478092475607, −8.129330855658595413732290958303, −6.77150262694981692716744982441, −5.82668293451764213232150375222, −4.24726335110309453798091096812, −3.80148212733984672807188260387, −1.77816137600723973628258470902, −0.913089449743657866476173567050, 1.50933413439452320662369104544, 2.98467005044535526292192763702, 4.17398487748319403129011284692, 5.96343645836205416414850622828, 6.40459633640473073233612379076, 7.12177555772978492401597610963, 8.065944647650922968091021917823, 9.081730075464808921460454232362, 9.777219200264224716084950369008, 10.65198479836826080420293394362

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