Properties

Label 2-28-28.11-c2-0-2
Degree $2$
Conductor $28$
Sign $0.938 - 0.345i$
Analytic cond. $0.762944$
Root an. cond. $0.873467$
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  + (1.78 + 0.907i)2-s + (−1.63 − 0.941i)3-s + (2.35 + 3.23i)4-s + (−1.12 − 1.95i)5-s + (−2.05 − 3.15i)6-s + (−6.84 − 1.47i)7-s + (1.25 + 7.90i)8-s + (−2.72 − 4.72i)9-s + (−0.236 − 4.49i)10-s + (8.47 + 4.89i)11-s + (−0.788 − 7.48i)12-s + 7.96·13-s + (−10.8 − 8.84i)14-s + 4.24i·15-s + (−4.94 + 15.2i)16-s + (−13.1 + 22.8i)17-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (−0.543 − 0.313i)3-s + (0.587 + 0.808i)4-s + (−0.225 − 0.390i)5-s + (−0.341 − 0.526i)6-s + (−0.977 − 0.210i)7-s + (0.156 + 0.987i)8-s + (−0.303 − 0.525i)9-s + (−0.0236 − 0.449i)10-s + (0.770 + 0.444i)11-s + (−0.0656 − 0.624i)12-s + 0.612·13-s + (−0.775 − 0.631i)14-s + 0.282i·15-s + (−0.308 + 0.951i)16-s + (−0.775 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18800 + 0.211687i\)
\(L(\frac12)\) \(\approx\) \(1.18800 + 0.211687i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78 - 0.907i)T \)
7 \( 1 + (6.84 + 1.47i)T \)
good3 \( 1 + (1.63 + 0.941i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.12 + 1.95i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.47 - 4.89i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 7.96T + 169T^{2} \)
17 \( 1 + (13.1 - 22.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-21.1 + 12.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.55 + 2.05i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 12.3T + 841T^{2} \)
31 \( 1 + (44.2 + 25.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (16.7 + 29.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 31.2T + 1.68e3T^{2} \)
43 \( 1 + 21.4iT - 1.84e3T^{2} \)
47 \( 1 + (39.4 - 22.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (7.90 - 13.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-54.5 - 31.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 - 32.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-28.1 - 16.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 16.1iT - 5.04e3T^{2} \)
73 \( 1 + (9.53 - 16.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-4.94 + 2.85i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 37.5iT - 6.88e3T^{2} \)
89 \( 1 + (-4.70 - 8.14i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 64.3T + 9.40e3T^{2} \)
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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

−16.87185252598173633591213492032, −15.89931594520871107871073439488, −14.67850600913582241634857092538, −13.16377004751719817656538537692, −12.38612026771062584789291332614, −11.17568967670975871891237176434, −8.981700368203798350655950717945, −7.00686308287124826930962537377, −5.92325574568361759109262127622, −3.86036894579724778449764873523, 3.37221915629247237289199890171, 5.39714044444212962364200555516, 6.80682284291695431626867815710, 9.463018397648893890595316354249, 10.95791888622747380852809478126, 11.74240327732765611184356371105, 13.24981845140533146799896168097, 14.31385392564539697176139462577, 15.84289669738925582938283383252, 16.44912860334505679418797388184

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