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Properties

Label 1-7e2-49.38-r1-0-0
Degree $1$
Conductor $49$
Sign $0.983 - 0.180i$
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  + (0.0747 − 0.997i)2-s + (−0.955 + 0.294i)3-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (0.733 − 0.680i)10-s + (0.826 + 0.563i)11-s + (0.988 − 0.149i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (−0.955 + 0.294i)3-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.826 − 0.563i)9-s + (0.733 − 0.680i)10-s + (0.826 + 0.563i)11-s + (0.988 − 0.149i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224545967 - 0.1115303703i\)
\(L(\frac12)\) \(\approx\) \(1.224545967 - 0.1115303703i\)
\(L(1)\) \(\approx\) \(0.9343417106 - 0.1862689289i\)
\(L(1)\) \(\approx\) \(0.9343417106 - 0.1862689289i\)

Euler product

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

−33.361704973107371942737630011763, −32.97537447269943777564872838771, −31.56811731692887178949143032334, −30.08187873257165039674399961179, −28.81264671953661790899085930116, −27.83650416930045244630584581182, −26.62566597142443453801502847651, −24.93487308526028692083579777913, −24.54327214674822111460127451994, −23.18233573918589053932002002184, −22.22837585595616822395889456469, −21.00345592974130548581083681848, −18.826271421710687451117483085963, −17.79194884667855385937392695735, −16.719922482472074704461484071507, −16.05921874728648926480823452932, −14.076196751851075745574638695934, −13.13391644033011175923630933284, −11.72630019830625098428630029537, −9.81806869164643058880341452901, −8.46414925859605441280988988506, −6.65873448928427922005854381711, −5.77097882608939115213710085004, −4.42660324299413679958922325730, −0.97597089274411999879268191015, 1.45378595458592628042566355923, 3.547511427272558060965051894309, 5.20392722253883702549199893755, 6.608402331282410613183258878644, 9.1518863387757853773006340264, 10.365804532313035264371729231386, 11.20902121605756938547440693166, 12.5485795625077466498980137443, 13.86092645752287761962658540342, 15.3726182302017012851911800054, 17.42060934601906181722439181915, 17.849206140318418130033953864174, 19.29912635284425535592784155638, 20.8331352262538071521103808405, 21.91029720225726008923700007420, 22.5666136633397212088936224044, 23.71987700023360750605157149259, 25.64898386770445378982676993667, 26.97004877078476032448844031790, 28.048352434846718514475689426416, 28.90490394001109400553876715290, 30.00832675411709425523304630790, 30.72348338987328022296450427523, 32.71969464835422910103913214241, 33.123721488488655439214626002519

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