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Properties

Label 1-7e2-49.39-r0-0-0
Degree $1$
Conductor $49$
Sign $0.991 - 0.127i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
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  + (0.955 + 0.294i)2-s + (0.365 − 0.930i)3-s + (0.826 + 0.563i)4-s + (−0.988 + 0.149i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.988 − 0.149i)10-s + (−0.733 + 0.680i)11-s + (0.826 − 0.563i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (0.365 − 0.930i)3-s + (0.826 + 0.563i)4-s + (−0.988 + 0.149i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.988 − 0.149i)10-s + (−0.733 + 0.680i)11-s + (0.826 − 0.563i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330637733 - 0.08542977694i\)
\(L(\frac12)\) \(\approx\) \(1.330637733 - 0.08542977694i\)
\(L(1)\) \(\approx\) \(1.497376326 - 0.05725960253i\)
\(L(1)\) \(\approx\) \(1.497376326 - 0.05725960253i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + T \)
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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

−33.70574318892804685853862949219, −32.34501179660431471401620203777, −31.58921922952435982852188521085, −30.940388947339481295276986115702, −29.330513938849831961941023226205, −28.13310395481958255798422796515, −27.03605971720192340642774216215, −25.7888013412687004353388984857, −24.20962632930678948764400427522, −23.28289606707130084029019993951, −21.9992314781284342786919297998, −21.07309241848162123565790824117, −19.97545657285596008758833549878, −19.01832151244170497590172326435, −16.43110541367687626713917002758, −15.72087356483794533645171396174, −14.59928258722468310821515956400, −13.360778851645792299025170619082, −11.672136512620887195363444365045, −10.831310679545591426563683962569, −9.126378929569324488787221386787, −7.37089748447892913034812335055, −5.2691573648571203008934620832, −4.12766929000533809832009000494, −2.850066632833490143980450551563, 2.47447128123156755365308935188, 3.998925599507717937789965092641, 5.90283386994892023595712208666, 7.439738180433075531259372471918, 8.12880084662476751108962297903, 10.83643171398057255273269501579, 12.38243410543024528483521183331, 12.90441522736308064912510543037, 14.61942841624935926472337733664, 15.3108302277856568318467021245, 16.95572976650942287513041571164, 18.52220622113332790883986972558, 19.854610007863292780674076329135, 20.720027424964980001405825136866, 22.59185625325205472918133634408, 23.37208599888442852978424003689, 24.29733773382151664454051526336, 25.43298148456378234145277402903, 26.48356823001286240713795612857, 28.297584695005528973526270020285, 29.74375058022985343423799112878, 30.58527771086907849952575903195, 31.40827476801357759521553955370, 32.30204312631567978726805372452, 33.93499038174627916039788289045

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