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Properties

Label 1-7e2-49.26-r1-0-0
Degree $1$
Conductor $49$
Sign $0.212 + 0.977i$
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  + (−0.988 − 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.0747 − 0.997i)5-s + (0.900 − 0.433i)6-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.0747 + 0.997i)10-s + (0.365 + 0.930i)11-s + (−0.955 + 0.294i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (0.826 + 0.563i)16-s + (0.733 + 0.680i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.0747 − 0.997i)5-s + (0.900 − 0.433i)6-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.0747 + 0.997i)10-s + (0.365 + 0.930i)11-s + (−0.955 + 0.294i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (0.826 + 0.563i)16-s + (0.733 + 0.680i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4526374945 + 0.3649396927i\)
\(L(\frac12)\) \(\approx\) \(0.4526374945 + 0.3649396927i\)
\(L(1)\) \(\approx\) \(0.5332979184 + 0.1017282858i\)
\(L(1)\) \(\approx\) \(0.5332979184 + 0.1017282858i\)

Euler product

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

−33.97399038148921317611032566564, −32.47129932748964562076157743388, −30.310205154609299272898627355096, −29.80665760024328432961035215962, −28.69947749859058258261246115205, −27.4439403651795772559763758470, −26.61253748374256932721496097459, −25.15878327367744369008123059169, −24.21670466719076062328618159778, −22.866111362017769199937239164983, −21.71563767876129177767051326878, −19.797682429616514590023757908655, −18.75908491162643972762062125610, −17.94859119417476904131241651892, −16.83283482842459197809184017565, −15.58736908400939826849537251926, −14.02126227293012270124399342984, −12.0181725750198338252462910458, −11.05515737358297221188709379746, −9.94747180657501843422216664260, −7.956986686952102998766231152754, −6.881279901881213725681474492565, −5.70735965166269706776401988443, −2.69442374460682321622145445074, −0.55994748654366490036597736261, 1.408585486354061211906783149012, 4.15260051906404845342471831553, 5.86024537608356784768719955870, 7.56429324856248070160313120345, 9.26633470280232120801367426568, 10.06125559903872400624944769196, 11.73826708443363893284862331561, 12.42111780073960012934846497126, 14.97949070631552482548006683371, 16.36905864873284539631381864573, 16.9440305267751633562005046288, 18.089853661694339894148606740815, 19.682777300469970552331404301138, 20.75790658628658318274808729945, 21.774430277321952624927913262099, 23.457709502767680430148608937549, 24.58497709080183659734125566438, 25.92765729252531665006231176550, 27.244490583313927299069584812705, 28.0383683469198228658006866716, 28.77962747920451630743785053817, 29.87798804536844025924147808263, 31.607980001208129352820864358752, 33.05540027285322622745576266029, 33.81673563514683632964833502994

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