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Properties

Label 2-2e7-16.13-c1-0-0
Degree $2$
Conductor $128$
Sign $0.923 - 0.382i$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)3-s + (1 − i)5-s + 2i·7-s − i·9-s + (−1 + i)11-s + (1 + i)13-s + 2·15-s − 2·17-s + (−3 − 3i)19-s + (−2 + 2i)21-s − 6i·23-s + 3i·25-s + (4 − 4i)27-s + (−3 − 3i)29-s − 8·31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (0.447 − 0.447i)5-s + 0.755i·7-s − 0.333i·9-s + (−0.301 + 0.301i)11-s + (0.277 + 0.277i)13-s + 0.516·15-s − 0.485·17-s + (−0.688 − 0.688i)19-s + (−0.436 + 0.436i)21-s − 1.25i·23-s + 0.600i·25-s + (0.769 − 0.769i)27-s + (−0.557 − 0.557i)29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27034 + 0.252686i\)
\(L(\frac12)\) \(\approx\) \(1.27034 + 0.252686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-3 + 3i)T - 59iT^{2} \)
61 \( 1 + (-9 - 9i)T + 61iT^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-1 - i)T + 83iT^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−13.33881577745211123614841678385, −12.55925731448152345563002571246, −11.33384213519873814057417509884, −10.07539591993287036465718180163, −9.066967263146620248958445397648, −8.560483771430705131742639013462, −6.78874395396345968355713133247, −5.43719444758836885544966933020, −4.10190629670526837288352010455, −2.40795688562533763666136812839, 2.03247494786444624826600944225, 3.66413976188351834094233483200, 5.52935668894085878039807304958, 6.93434372952847118207164926085, 7.80829372269644100130931618851, 8.932410934063420078611538180706, 10.35542828808534380960988333069, 10.98197856559606242788936714678, 12.55950530458549824198814810685, 13.50457575276406337362419000987

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