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Properties

Label 2-28-4.3-c2-0-3
Degree $2$
Conductor $28$
Sign $0.965 + 0.260i$
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  + (1.58 − 1.21i)2-s + 2.98i·3-s + (1.04 − 3.86i)4-s − 6.80·5-s + (3.62 + 4.73i)6-s − 2.64i·7-s + (−3.04 − 7.39i)8-s + 0.0914·9-s + (−10.8 + 8.27i)10-s + 14.3i·11-s + (11.5 + 3.11i)12-s + 4.62·13-s + (−3.21 − 4.20i)14-s − 20.3i·15-s + (−13.8 − 8.04i)16-s + 11.6·17-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + 0.994i·3-s + (0.260 − 0.965i)4-s − 1.36·5-s + (0.604 + 0.789i)6-s − 0.377i·7-s + (−0.380 − 0.924i)8-s + 0.0101·9-s + (−1.08 + 0.827i)10-s + 1.30i·11-s + (0.960 + 0.259i)12-s + 0.355·13-s + (−0.229 − 0.300i)14-s − 1.35i·15-s + (−0.864 − 0.503i)16-s + 0.682·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.965 + 0.260i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.965 + 0.260i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20205 - 0.159325i\)
\(L(\frac12)\) \(\approx\) \(1.20205 - 0.159325i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.58 + 1.21i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 2.98iT - 9T^{2} \)
5 \( 1 + 6.80T + 25T^{2} \)
11 \( 1 - 14.3iT - 121T^{2} \)
13 \( 1 - 4.62T + 169T^{2} \)
17 \( 1 - 11.6T + 289T^{2} \)
19 \( 1 + 22.4iT - 361T^{2} \)
23 \( 1 + 0.853iT - 529T^{2} \)
29 \( 1 + 42.8T + 841T^{2} \)
31 \( 1 - 1.54iT - 961T^{2} \)
37 \( 1 - 45.0T + 1.36e3T^{2} \)
41 \( 1 + 36.6T + 1.68e3T^{2} \)
43 \( 1 - 27.9iT - 1.84e3T^{2} \)
47 \( 1 + 42.1iT - 2.20e3T^{2} \)
53 \( 1 - 43.4T + 2.80e3T^{2} \)
59 \( 1 - 53.9iT - 3.48e3T^{2} \)
61 \( 1 + 15.9T + 3.72e3T^{2} \)
67 \( 1 + 91.9iT - 4.48e3T^{2} \)
71 \( 1 + 16.3iT - 5.04e3T^{2} \)
73 \( 1 - 9.58T + 5.32e3T^{2} \)
79 \( 1 - 56.9iT - 6.24e3T^{2} \)
83 \( 1 + 14.3iT - 6.88e3T^{2} \)
89 \( 1 - 100.T + 7.92e3T^{2} \)
97 \( 1 - 68.2T + 9.40e3T^{2} \)
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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

−16.48460997971103198381771315875, −15.35620001394200219990857600530, −14.91490985975737554058128594623, −13.08758457923372806184417731580, −11.82077007454108685194156982963, −10.75297938948660470373669580677, −9.536751602142922064651127250167, −7.27738920409000896145478682185, −4.76305627648873486823087021906, −3.75144285019093332948504884214, 3.66998147039404878475458260250, 5.94525890427454232628894389526, 7.51144979732976666144419793992, 8.328026275418219552674211718193, 11.37896431897297833894547378600, 12.24411359807794365211778655605, 13.31047674954516558387159180193, 14.60783569829594830086338971914, 15.82326885786318822509208183701, 16.68609256943143677146207316253

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