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Properties

Label 2-1071-17.13-c1-0-1
Degree $2$
Conductor $1071$
Sign $0.910 - 0.414i$
Analytic cond. $8.55197$
Root an. cond. $2.92437$
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  − 2.05i·2-s − 2.21·4-s + (−1.82 − 1.82i)5-s + (−0.707 + 0.707i)7-s + 0.439i·8-s + (−3.73 + 3.73i)10-s + (−3.22 + 3.22i)11-s + 0.539·13-s + (1.45 + 1.45i)14-s − 3.52·16-s + (1.23 + 3.93i)17-s + 2.26i·19-s + (4.03 + 4.03i)20-s + (6.61 + 6.61i)22-s + (0.756 − 0.756i)23-s + ⋯
L(s)  = 1  − 1.45i·2-s − 1.10·4-s + (−0.814 − 0.814i)5-s + (−0.267 + 0.267i)7-s + 0.155i·8-s + (−1.18 + 1.18i)10-s + (−0.971 + 0.971i)11-s + 0.149·13-s + (0.387 + 0.387i)14-s − 0.881·16-s + (0.300 + 0.953i)17-s + 0.518i·19-s + (0.901 + 0.901i)20-s + (1.41 + 1.41i)22-s + (0.157 − 0.157i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3463245899\)
\(L(\frac12)\) \(\approx\) \(0.3463245899\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-1.23 - 3.93i)T \)
good2 \( 1 + 2.05iT - 2T^{2} \)
5 \( 1 + (1.82 + 1.82i)T + 5iT^{2} \)
11 \( 1 + (3.22 - 3.22i)T - 11iT^{2} \)
13 \( 1 - 0.539T + 13T^{2} \)
19 \( 1 - 2.26iT - 19T^{2} \)
23 \( 1 + (-0.756 + 0.756i)T - 23iT^{2} \)
29 \( 1 + (-1.56 - 1.56i)T + 29iT^{2} \)
31 \( 1 + (3.92 + 3.92i)T + 31iT^{2} \)
37 \( 1 + (-1.70 - 1.70i)T + 37iT^{2} \)
41 \( 1 + (3.78 - 3.78i)T - 41iT^{2} \)
43 \( 1 - 3.43iT - 43T^{2} \)
47 \( 1 + 6.05T + 47T^{2} \)
53 \( 1 - 9.01iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (-2.48 + 2.48i)T - 61iT^{2} \)
67 \( 1 + 2.85T + 67T^{2} \)
71 \( 1 + (-6.24 - 6.24i)T + 71iT^{2} \)
73 \( 1 + (-11.2 - 11.2i)T + 73iT^{2} \)
79 \( 1 + (3.24 - 3.24i)T - 79iT^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (-5.81 - 5.81i)T + 97iT^{2} \)
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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

−10.04178230602522956809816930422, −9.429522247831689807281142595893, −8.391393002700575358831980406034, −7.81144899093010271770488160761, −6.58630034051254992174752005713, −5.26732893158456166078724146996, −4.40159604949025959466634201472, −3.62948233990304067040626644523, −2.51878399967438198337469567018, −1.38131909054823229179317987650, 0.15851429606278255180813262356, 2.75080555628048548923356641871, 3.66023965566887912575649177855, 4.96570086076680947397128600124, 5.67145261789784840877518019384, 6.74651316954018888565766902513, 7.20804306286820096211483492986, 7.953792970802178176420549144792, 8.617592691228112599008244911129, 9.621221450736845905874598739165

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