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Properties

Label 1-1649-1649.38-r1-0-0
Degree $1$
Conductor $1649$
Sign $0.974 + 0.226i$
Analytic cond. $177.209$
Root an. cond. $177.209$
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  + (0.130 + 0.991i)2-s + (−0.793 − 0.608i)3-s + (−0.965 + 0.258i)4-s + (0.946 − 0.321i)5-s + (0.5 − 0.866i)6-s + (−0.751 + 0.659i)7-s + (−0.382 − 0.923i)8-s + (0.258 + 0.965i)9-s + (0.442 + 0.896i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (0.946 − 0.321i)13-s + (−0.751 − 0.659i)14-s + (−0.946 − 0.321i)15-s + (0.866 − 0.5i)16-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.793 − 0.608i)3-s + (−0.965 + 0.258i)4-s + (0.946 − 0.321i)5-s + (0.5 − 0.866i)6-s + (−0.751 + 0.659i)7-s + (−0.382 − 0.923i)8-s + (0.258 + 0.965i)9-s + (0.442 + 0.896i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (0.946 − 0.321i)13-s + (−0.751 − 0.659i)14-s + (−0.946 − 0.321i)15-s + (0.866 − 0.5i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1649\)    =    \(17 \cdot 97\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(177.209\)
Root analytic conductor: \(177.209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1649} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1649,\ (1:\ ),\ 0.974 + 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159307267 + 0.1330330844i\)
\(L(\frac12)\) \(\approx\) \(1.159307267 + 0.1330330844i\)
\(L(1)\) \(\approx\) \(0.7660210111 + 0.2820923561i\)
\(L(1)\) \(\approx\) \(0.7660210111 + 0.2820923561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.130 + 0.991i)T \)
3 \( 1 + (-0.793 - 0.608i)T \)
5 \( 1 + (0.946 - 0.321i)T \)
7 \( 1 + (-0.751 + 0.659i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (0.946 - 0.321i)T \)
19 \( 1 + (-0.980 - 0.195i)T \)
23 \( 1 + (-0.997 + 0.0654i)T \)
29 \( 1 + (-0.442 + 0.896i)T \)
31 \( 1 + (0.608 - 0.793i)T \)
37 \( 1 + (0.0654 - 0.997i)T \)
41 \( 1 + (0.896 + 0.442i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.991 + 0.130i)T \)
59 \( 1 + (0.0654 - 0.997i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (0.896 - 0.442i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.659 - 0.751i)T \)
89 \( 1 + (-0.382 - 0.923i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.54259496558578503761449735471, −19.33774106597676087202941998499, −18.79917485752309788374433454351, −17.9850761348561701792558725900, −17.30551812922363491608383786934, −16.625780882577444066013058864803, −15.899434474151043365414951366047, −14.811943094311021308344278737735, −13.83653491334313050923993718874, −13.48275092557552643329994550018, −12.62355002198582789289341860734, −11.738102438438715791742114094368, −10.78288366476810033134360437054, −10.59053668778235500395511678219, −9.76888233219113658159761736087, −9.14242054927529376847143332174, −8.190931056339389734729478179834, −6.481491186711131165159585764360, −6.1835343076674684258697261933, −5.34684275194690561927053256885, −4.23294446866423039338520705194, −3.64338389973618939893972877286, −2.7764657147384211858667972254, −1.55111850464222128799973096358, −0.6244761339032212269512727273, 0.36309651943572630252110588162, 1.5944900322674645010029937629, 2.54982702406929467883403823918, 3.9889093811769374617909678507, 4.96038701782592902307083550903, 5.671488936457002515490921597746, 6.29511669697792961802867957234, 6.733096743617063097434257839893, 7.85165762862883236992827017596, 8.60430509396588565681388459682, 9.56358488646792385893309644421, 10.10093297984382594756600482906, 11.21485973299426892137658048257, 12.501342824159681326856059146827, 12.762277000880504391302824317434, 13.31579062138975718304422038723, 14.23031812321172171721134266209, 15.12880043092123852810941838128, 16.04161032006617249363941910676, 16.42560554905845391619149722639, 17.40659018509420225887980258050, 17.838929970407453992629000842937, 18.40710461605175806241046082746, 19.14673404130515883459957011220, 20.235296544204132382098179029379

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