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Properties

Label 2-1456-1456.909-c0-0-1
Degree $2$
Conductor $1456$
Sign $0.382 - 0.923i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1 − i)5-s − 1.00i·6-s + i·7-s + (0.707 + 0.707i)8-s + 1.41i·10-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1 − i)5-s − 1.00i·6-s + i·7-s + (0.707 + 0.707i)8-s + 1.41i·10-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7797606293\)
\(L(\frac12)\) \(\approx\) \(0.7797606293\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
5 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.655590437792111463806365418362, −9.090566698173040373034112088806, −8.604833347235443264358361698918, −7.48640042505354076597263905209, −6.32727962517098228032946088299, −5.76928669328327016899659149965, −5.03721784641461589968962507394, −4.57898819383636527742667408396, −2.50492776890058240779525438810, −1.23825314178294678850141543441, 1.14811745571244708804677797681, 1.98202892081072260813157070856, 3.43097086748211374253963680676, 4.06985775652604471544933729588, 5.96076195207384291337354769225, 6.34813064077060578883590471338, 7.07396119869336432719803419029, 7.88089408723271353006094624359, 8.992788111473966454076039803820, 9.679039374501138807911721787006

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