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Properties

Label 2-1028-1028.839-c0-0-0
Degree $2$
Conductor $1028$
Sign $-0.762 - 0.647i$
Analytic cond. $0.513038$
Root an. cond. $0.716267$
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  + i·2-s − 4-s + (0.577 + 1.90i)5-s − i·8-s + (0.555 − 0.831i)9-s + (−1.90 + 0.577i)10-s + (0.324 + 1.63i)13-s + 16-s + (−0.275 − 0.275i)17-s + (0.831 + 0.555i)18-s + (−0.577 − 1.90i)20-s + (−2.46 + 1.64i)25-s + (−1.63 + 0.324i)26-s + (−0.785 − 1.17i)29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.577 + 1.90i)5-s − i·8-s + (0.555 − 0.831i)9-s + (−1.90 + 0.577i)10-s + (0.324 + 1.63i)13-s + 16-s + (−0.275 − 0.275i)17-s + (0.831 + 0.555i)18-s + (−0.577 − 1.90i)20-s + (−2.46 + 1.64i)25-s + (−1.63 + 0.324i)26-s + (−0.785 − 1.17i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1028\)    =    \(2^{2} \cdot 257\)
Sign: $-0.762 - 0.647i$
Analytic conductor: \(0.513038\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1028} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1028,\ (\ :0),\ -0.762 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.022844408\)
\(L(\frac12)\) \(\approx\) \(1.022844408\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
257 \( 1 + T \)
good3 \( 1 + (-0.555 + 0.831i)T^{2} \)
5 \( 1 + (-0.577 - 1.90i)T + (-0.831 + 0.555i)T^{2} \)
7 \( 1 + (0.195 + 0.980i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
19 \( 1 + (0.980 - 0.195i)T^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.785 + 1.17i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-1.53 + 1.26i)T + (0.195 - 0.980i)T^{2} \)
41 \( 1 + (1.90 + 0.187i)T + (0.980 + 0.195i)T^{2} \)
43 \( 1 + (0.555 + 0.831i)T^{2} \)
47 \( 1 + (0.980 - 0.195i)T^{2} \)
53 \( 1 + (-0.187 - 0.0569i)T + (0.831 + 0.555i)T^{2} \)
59 \( 1 + (0.923 + 0.382i)T^{2} \)
61 \( 1 + (-0.360 - 1.81i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.980 + 0.195i)T^{2} \)
73 \( 1 + (-1.02 - 0.425i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.555 + 0.831i)T^{2} \)
89 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
97 \( 1 + (-0.902 + 0.273i)T + (0.831 - 0.555i)T^{2} \)
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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

−10.13210270637338497216095222119, −9.632466602307902332998610296137, −8.909768913784380613368796337137, −7.56903192263160573143309011265, −6.92230598491789434465135773606, −6.46256180440686522229142613297, −5.74779279750816086626993159067, −4.23271005212496875278273531961, −3.52778337718443819285208089910, −2.08731970455126486790200488525, 1.07657613422300720616541096577, 2.02202125426084213664154871796, 3.48405024004461388835474436599, 4.76133481010756717716486626353, 5.06419800085307463995320478223, 6.00388005766494265634425930871, 7.88046450978349566046509844812, 8.291420273823705977474168743446, 9.124064419878304620754817586965, 9.900507759672766213565871961685

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