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Properties

Label 2-684-76.51-c1-0-37
Degree $2$
Conductor $684$
Sign $0.823 + 0.566i$
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  + (1.37 + 0.330i)2-s + (1.78 + 0.907i)4-s + (−0.432 − 2.45i)5-s + (−1.95 − 1.13i)7-s + (2.15 + 1.83i)8-s + (0.214 − 3.51i)10-s + (3.50 − 2.02i)11-s + (−0.674 − 1.85i)13-s + (−2.32 − 2.20i)14-s + (2.35 + 3.23i)16-s + (4.59 − 3.85i)17-s + (−1.31 + 4.15i)19-s + (1.45 − 4.76i)20-s + (5.49 − 1.62i)22-s + (−2.40 − 0.424i)23-s + ⋯
L(s)  = 1  + (0.972 + 0.233i)2-s + (0.891 + 0.453i)4-s + (−0.193 − 1.09i)5-s + (−0.740 − 0.427i)7-s + (0.760 + 0.649i)8-s + (0.0679 − 1.11i)10-s + (1.05 − 0.610i)11-s + (−0.187 − 0.514i)13-s + (−0.620 − 0.588i)14-s + (0.587 + 0.809i)16-s + (1.11 − 0.934i)17-s + (−0.300 + 0.953i)19-s + (0.325 − 1.06i)20-s + (1.17 − 0.347i)22-s + (−0.501 − 0.0884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50233 - 0.777778i\)
\(L(\frac12)\) \(\approx\) \(2.50233 - 0.777778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.330i)T \)
3 \( 1 \)
19 \( 1 + (1.31 - 4.15i)T \)
good5 \( 1 + (0.432 + 2.45i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.95 + 1.13i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.50 + 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.674 + 1.85i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.59 + 3.85i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (2.40 + 0.424i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.27 + 7.47i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.39 - 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.46iT - 37T^{2} \)
41 \( 1 + (3.98 - 10.9i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.50 - 0.265i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.929 - 1.10i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (2.26 + 0.399i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (8.73 - 7.33i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.80 - 10.2i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.48 - 7.12i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.05 + 5.97i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (3.23 + 1.17i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-5.12 - 1.86i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.35 - 3.66i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.28 - 11.7i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.13 + 3.73i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.40057917598416150954211416538, −9.606703775414561407152073651472, −8.441436906265965421332080460046, −7.76240936992972687943653372610, −6.60852016080471628134277448899, −5.86511577345450817043449468781, −4.85563899601223094813808227119, −3.93152679593138519291517479386, −3.03702278642187665144624368577, −1.13803924226716431954011901280, 1.88647981577987863043993134909, 3.11970222343621710630472409531, 3.81443045083556257687948486499, 5.02589140048923278733986904978, 6.34119406277220387905675864230, 6.64503732127797172802952932107, 7.56913266490325265332445621763, 9.046914033031947361998834030557, 9.966958005200814878271635225770, 10.67208121791005285937499865957

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