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Properties

Label 1-7e2-49.12-r1-0-0
Degree $1$
Conductor $49$
Sign $-0.952 + 0.304i$
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  + (0.365 − 0.930i)2-s + (−0.0747 + 0.997i)3-s + (−0.733 − 0.680i)4-s + (−0.826 − 0.563i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.826 + 0.563i)10-s + (−0.988 + 0.149i)11-s + (0.733 − 0.680i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.0747 + 0.997i)3-s + (−0.733 − 0.680i)4-s + (−0.826 − 0.563i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.826 + 0.563i)10-s + (−0.988 + 0.149i)11-s + (0.733 − 0.680i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.952 + 0.304i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ -0.952 + 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04478765454 - 0.2867429705i\)
\(L(\frac12)\) \(\approx\) \(-0.04478765454 - 0.2867429705i\)
\(L(1)\) \(\approx\) \(0.6299325149 - 0.2733127970i\)
\(L(1)\) \(\approx\) \(0.6299325149 - 0.2733127970i\)

Euler product

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

−34.425103294481720862852571863887, −33.27946418283174752495980237990, −31.4027600362506271150874962529, −31.20345640500677529040080080220, −29.90922578920908912027717357305, −28.48829154281247660858387551275, −26.7652693046195620356961412229, −26.070794865864016378713250777325, −24.59130696927847982327819842932, −23.776819139463192383600305987180, −22.96208758133207714226406267220, −21.70719004381775877270077214856, −19.69023749653600805770813316781, −18.55664964026700591629174374166, −17.56586009972557578607644621417, −16.09194118077572751125847049995, −14.89356098798165439767292385166, −13.6712493802738866935807753956, −12.518820437268304944858698839897, −11.20052101297036223109297393650, −8.75231749872514304311631403287, −7.46012940228611361444557203134, −6.704950914926412670963557939494, −4.945545814423314338250387884439, −2.9556282127642816249688361579, 0.146418463546257642621780970056, 2.92075589537154535580161975154, 4.379211240550190963198196202542, 5.360312980669471709452640618995, 8.1844274667622825309211682669, 9.59869405259367974219006399114, 10.775549997814454111224246391327, 11.917306693878318363746403008747, 13.199942419857201649616020290960, 14.91483832022547657921202223613, 15.795159270168181227857287900827, 17.408830560682659913072300410283, 19.0591683416760588706718442985, 20.36731172380439103088745215556, 20.84164874470339515272733939406, 22.378697608194720723937008034469, 23.09792145148174390544516236859, 24.50036153995617718317889793658, 26.57356602853647273088730302337, 27.3192639707573995948783059547, 28.35407481003819988028687912503, 29.218501663889418170373404857473, 31.00188085097270187969320032697, 31.63647060202183664644098469536, 32.57866844644756204056352999345

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