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Properties

Label 1-7e2-49.15-r0-0-0
Degree $1$
Conductor $49$
Sign $-0.761 - 0.648i$
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  + (−0.900 − 0.433i)2-s + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + 18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ -0.761 - 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1570800948 - 0.4268373620i\)
\(L(\frac12)\) \(\approx\) \(0.1570800948 - 0.4268373620i\)
\(L(1)\) \(\approx\) \(0.4366845057 - 0.3825441013i\)
\(L(1)\) \(\approx\) \(0.4366845057 - 0.3825441013i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
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

−34.20071135563324623740035470969, −33.5317886314063611596578998506, −32.34096247146441889670034148592, −30.89847796934645522520167015232, −29.23564306114711642774304216754, −28.346722925327816024040611171110, −27.02768989486612229440781973286, −26.484949728525398833522118732750, −25.49667130536254073774493272494, −23.77540328673930719136339140191, −22.68721944438723484399487505576, −21.32266906771563983246996079616, −19.98870248429834940166339868895, −18.71368516128555029382072606383, −17.55185138112648092126002908951, −16.33830144923573849041085251373, −15.215173683105178221869488634361, −14.4320396594566081730135319926, −11.7631775228370578818945767035, −10.48726256328192917339560235123, −9.78619198348865543619779416373, −8.06985933497139805237288880742, −6.64776176162004323808696295551, −5.044400320231324786207707825397, −2.83973610442772228754210029031, 0.871228361333598126613073440567, 2.76062525074196651097524011736, 5.374016248721491520259061091213, 7.387924779242156129839925566160, 8.21682653824405354047776723879, 9.72873650042050863621263757421, 11.46914214076433103532338447528, 12.38588567579592584913567933935, 13.49066419367160554525274440230, 15.83567054342101594863191466743, 16.966999506901093452462007399803, 17.98143934269544131113897285764, 19.1621208495271326287567150667, 20.09854402443800123057272093322, 21.26674951413903569932902683326, 23.04926476654701159201524885834, 24.46278817166579249104830290813, 25.073941949109800427360976231025, 26.62783909877850526921322068051, 27.838628417971457212340872253529, 28.92972136445549302897187940370, 29.50987986808113683545497197625, 30.91614338290827164799929050346, 31.90668780734970529604202540631, 33.89774573435192908608024634253

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