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Properties

Label 2-1028-1028.907-c0-0-0
Degree $2$
Conductor $1028$
Sign $0.662 + 0.749i$
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 + (−0.368 + 0.448i)5-s + i·8-s + (0.980 + 0.195i)9-s + (0.448 + 0.368i)10-s + (0.216 + 0.324i)13-s + 16-s + (0.785 − 0.785i)17-s + (0.195 − 0.980i)18-s + (0.368 − 0.448i)20-s + (0.129 + 0.650i)25-s + (0.324 − 0.216i)26-s + (1.38 − 0.275i)29-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.368 + 0.448i)5-s + i·8-s + (0.980 + 0.195i)9-s + (0.448 + 0.368i)10-s + (0.216 + 0.324i)13-s + 16-s + (0.785 − 0.785i)17-s + (0.195 − 0.980i)18-s + (0.368 − 0.448i)20-s + (0.129 + 0.650i)25-s + (0.324 − 0.216i)26-s + (1.38 − 0.275i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1028\)    =    \(2^{2} \cdot 257\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(0.513038\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1028} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1028,\ (\ :0),\ 0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9535850294\)
\(L(\frac12)\) \(\approx\) \(0.9535850294\)
\(L(1)\) 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 \( 1 + (-0.980 - 0.195i)T^{2} \)
5 \( 1 + (0.368 - 0.448i)T + (-0.195 - 0.980i)T^{2} \)
7 \( 1 + (0.555 + 0.831i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
17 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
19 \( 1 + (-0.831 + 0.555i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.512 - 0.273i)T + (0.555 - 0.831i)T^{2} \)
41 \( 1 + (-0.448 + 1.47i)T + (-0.831 - 0.555i)T^{2} \)
43 \( 1 + (0.980 - 0.195i)T^{2} \)
47 \( 1 + (-0.831 + 0.555i)T^{2} \)
53 \( 1 + (1.47 - 1.21i)T + (0.195 - 0.980i)T^{2} \)
59 \( 1 + (0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.425 - 0.636i)T + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.831 - 0.555i)T^{2} \)
73 \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.980 - 0.195i)T^{2} \)
89 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
97 \( 1 + (0.151 + 0.124i)T + (0.195 + 0.980i)T^{2} \)
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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

−10.11167941637607208348620059926, −9.481581977842810769409623177836, −8.516772840452491254031531162415, −7.60081222632316254781034802296, −6.81528609642138436472278272791, −5.46590857692400235798532969497, −4.53785579019211182800464160388, −3.66412918075816231827281868818, −2.68462711100802008411557846431, −1.33718969264319444772434139319, 1.21394696471838532946309467986, 3.35232173063528365131372662894, 4.33279426890560608646962687552, 5.03094488666880271425307474244, 6.17688113524168087259934647558, 6.81477499833755512908739883613, 7.983296733752818377761147704685, 8.203001067291492975309955301124, 9.361098434411238862965525944630, 10.00467804558299216548528505483

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