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Properties

Label 1-7e2-49.3-r1-0-0
Degree $1$
Conductor $49$
Sign $0.127 - 0.991i$
Analytic cond. $5.26578$
Root an. cond. $5.26578$
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  + (−0.733 − 0.680i)2-s + (0.988 + 0.149i)3-s + (0.0747 + 0.997i)4-s + (−0.365 − 0.930i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (−0.365 + 0.930i)10-s + (0.955 − 0.294i)11-s + (−0.0747 + 0.997i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 + 0.149i)16-s + (−0.826 − 0.563i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.988 + 0.149i)3-s + (0.0747 + 0.997i)4-s + (−0.365 − 0.930i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (−0.365 + 0.930i)10-s + (0.955 − 0.294i)11-s + (−0.0747 + 0.997i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 + 0.149i)16-s + (−0.826 − 0.563i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.127 - 0.991i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ 0.127 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068671029 - 0.9397275615i\)
\(L(\frac12)\) \(\approx\) \(1.068671029 - 0.9397275615i\)
\(L(1)\) \(\approx\) \(0.9583339522 - 0.4611601525i\)
\(L(1)\) \(\approx\) \(0.9583339522 - 0.4611601525i\)

Euler product

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

−33.74120653860952480037794741154, −32.93634403592937280749210100503, −31.50631392159398751874426341013, −30.52784512813408811523082229005, −29.20412442214439549263913657725, −27.57175495863451255755118290810, −26.64893427243697368346363118817, −25.87406034926536454134944033128, −24.80543345502714281373314346802, −23.68382088522433418680153932439, −22.25663428991473139677028461202, −20.4531728858376616208254639101, −19.227663909906532571154922090250, −18.67158767785496299037124527744, −17.17148951518147306663503171936, −15.56283504405165589493314370014, −14.71858862938459726337494978965, −13.73842375709841462780750706705, −11.498618796693628038571943231180, −9.93066799689361098357880952263, −8.79121891288189077374272955840, −7.4329222582039695035891962992, −6.48476961657871025558655003307, −3.93972925775712889760018875837, −1.87588640103161486310635060704, 1.08429970573353567979478609315, 3.004126893189794793638884791447, 4.442730142130269454535856856115, 7.34413238685977309801260308167, 8.68265276177954400082469776803, 9.3083903911732875575822287674, 11.00661021895721093999119707588, 12.51035253272093663354925120561, 13.56249502948809405078576557439, 15.43699245305702118380837213155, 16.57098243974708675414928949476, 18.00078902259586369728517784594, 19.51202192824979512123113039027, 20.078388782986246820207569064248, 21.04573532001405216359050198709, 22.37762932995809813064430935661, 24.511827715360102391509015471516, 25.209179746787339278124659786123, 26.67373548203036622360327919406, 27.42662320056560321661079601373, 28.49929643867283542601538567876, 29.933967999417902977318832158687, 30.87731471067510578528788982371, 31.95056537726300491188541437290, 32.97724441447080788354457695068

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