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Properties

Label 1-1649-1649.405-r0-0-0
Degree $1$
Conductor $1649$
Sign $0.000652 - 0.999i$
Analytic cond. $7.65792$
Root an. cond. $7.65792$
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  + (−0.793 + 0.608i)2-s + (−0.965 + 0.258i)3-s + (0.258 − 0.965i)4-s + (−0.0654 − 0.997i)5-s + (0.608 − 0.793i)6-s + (0.896 − 0.442i)7-s + (0.382 + 0.923i)8-s + (0.866 − 0.5i)9-s + (0.659 + 0.751i)10-s + (−0.5 − 0.866i)11-s + i·12-s + (−0.896 + 0.442i)13-s + (−0.442 + 0.896i)14-s + (0.321 + 0.946i)15-s + (−0.866 − 0.5i)16-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (−0.965 + 0.258i)3-s + (0.258 − 0.965i)4-s + (−0.0654 − 0.997i)5-s + (0.608 − 0.793i)6-s + (0.896 − 0.442i)7-s + (0.382 + 0.923i)8-s + (0.866 − 0.5i)9-s + (0.659 + 0.751i)10-s + (−0.5 − 0.866i)11-s + i·12-s + (−0.896 + 0.442i)13-s + (−0.442 + 0.896i)14-s + (0.321 + 0.946i)15-s + (−0.866 − 0.5i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000652 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000652 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1649\)    =    \(17 \cdot 97\)
Sign: $0.000652 - 0.999i$
Analytic conductor: \(7.65792\)
Root analytic conductor: \(7.65792\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1649} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1649,\ (0:\ ),\ 0.000652 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4808202621 - 0.4805067328i\)
\(L(\frac12)\) \(\approx\) \(0.4808202621 - 0.4805067328i\)
\(L(1)\) \(\approx\) \(0.5841351538 - 0.07310443506i\)
\(L(1)\) \(\approx\) \(0.5841351538 - 0.07310443506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.793 + 0.608i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.0654 - 0.997i)T \)
7 \( 1 + (0.896 - 0.442i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.896 + 0.442i)T \)
19 \( 1 + (0.980 - 0.195i)T \)
23 \( 1 + (0.442 - 0.896i)T \)
29 \( 1 + (-0.659 + 0.751i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.896 - 0.442i)T \)
41 \( 1 + (0.997 - 0.0654i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.608 + 0.793i)T \)
59 \( 1 + (-0.0654 - 0.997i)T \)
61 \( 1 + (0.991 + 0.130i)T \)
67 \( 1 + (-0.555 + 0.831i)T \)
71 \( 1 + (0.0654 - 0.997i)T \)
73 \( 1 + (0.130 - 0.991i)T \)
79 \( 1 - iT \)
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

−20.56782348751352637389491994113, −19.65395688312689926693086921963, −18.917388134716010255213033906783, −18.15853257261590641480865065954, −17.81295691810515053916372967964, −17.33765904760039084464649884548, −16.30706353437510063301354389470, −15.4305158203576187122039258242, −14.848159808219995745334467102, −13.65777022829383240379699982199, −12.71575706925775983494619540864, −12.02922721046835524504800778560, −11.40510697664687174615485930429, −10.87537717919590108346989367626, −9.99831392736599317140601013667, −9.5527898051001246785531083892, −8.09578007263023296111176402110, −7.421736844765319048244167093046, −7.11072253143687188594365875584, −5.80929964172249636527230085719, −5.023478122308007181461642299784, −4.013354031423882669138057851760, −2.74676995704534005834687991121, −2.11691057803210472213489089300, −1.10817030714271578683555601075, 0.48023169444655845734949706473, 1.11740776914876793068320899693, 2.27276289033723093159914574462, 4.07331813541065315590271805077, 4.88335852233184916815136091152, 5.3839985472284527935855723153, 6.13654352014512431292542800232, 7.39716812117499966227072286655, 7.68114911575453177987066685977, 8.872610345150207826231274584093, 9.37684812718170434568870795258, 10.36200557059942273372051665367, 11.066925659489885480635500796, 11.62774900853757214278349040307, 12.557796054387667414814557054673, 13.546251309789282351782737366884, 14.43050979942198049800807729812, 15.23221588240863720126292306881, 16.20766988129177581031325815583, 16.5014925094764185232431230214, 17.15439394945208142863824822627, 17.77321554694918098536624226378, 18.5148873929599570751138793728, 19.270382361898523953830109152572, 20.3819809757951630120157555455

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