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Properties

Label 2-2e7-32.29-c1-0-1
Degree $2$
Conductor $128$
Sign $0.987 - 0.159i$
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  + (2.27 + 0.943i)3-s + (−0.707 − 1.70i)5-s + (0.665 − 0.665i)7-s + (2.18 + 2.18i)9-s + (−3.69 + 1.52i)11-s + (−1.76 + 4.26i)13-s − 4.55i·15-s − 3.61i·17-s + (−0.194 + 0.470i)19-s + (2.14 − 0.887i)21-s + (1.33 + 1.33i)23-s + (1.12 − 1.12i)25-s + (0.0793 + 0.191i)27-s + (−5.73 − 2.37i)29-s − 1.17·31-s + ⋯
L(s)  = 1  + (1.31 + 0.544i)3-s + (−0.316 − 0.763i)5-s + (0.251 − 0.251i)7-s + (0.726 + 0.726i)9-s + (−1.11 + 0.461i)11-s + (−0.489 + 1.18i)13-s − 1.17i·15-s − 0.877i·17-s + (−0.0446 + 0.107i)19-s + (0.467 − 0.193i)21-s + (0.278 + 0.278i)23-s + (0.224 − 0.224i)25-s + (0.0152 + 0.0368i)27-s + (−1.06 − 0.441i)29-s − 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41976 + 0.113613i\)
\(L(\frac12)\) \(\approx\) \(1.41976 + 0.113613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.27 - 0.943i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.707 + 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.665 + 0.665i)T - 7iT^{2} \)
11 \( 1 + (3.69 - 1.52i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.76 - 4.26i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.61iT - 17T^{2} \)
19 \( 1 + (0.194 - 0.470i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.33 - 1.33i)T + 23iT^{2} \)
29 \( 1 + (5.73 + 2.37i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + (-0.510 - 1.23i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.66 - 1.66i)T + 41iT^{2} \)
43 \( 1 + (-2.54 + 1.05i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 + (-4.59 + 1.90i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.04 + 4.94i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-13.7 - 5.67i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.40 - 1.41i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.66 + 9.66i)T - 71iT^{2} \)
73 \( 1 + (7.55 + 7.55i)T + 73iT^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 + (4.82 - 11.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.43 - 5.43i)T - 89iT^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−13.54856420288544950332061181460, −12.53321849371569887414560457025, −11.30325143733274083816895989126, −9.888977027817116455909385165947, −9.172813123173065276186982103677, −8.170490793817523180871508398904, −7.26092375628937437266656310881, −5.04317257263903977451432476937, −4.06115715062294563523034326354, −2.41943147861075777724173835435, 2.44255553727107700784220926850, 3.42727945181068356969416345193, 5.50700713458614489203156009768, 7.20814024618736523841046655016, 7.943602970551193291262875724145, 8.782539645778989845530345802688, 10.25068770882725357798823081286, 11.13574657813235821860905367587, 12.74019228595558659227921996601, 13.22169308856242847703415634393

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