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Properties

Label 1-7e2-49.32-r0-0-0
Degree $1$
Conductor $49$
Sign $0.977 + 0.212i$
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  + (−0.988 + 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

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

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.977 + 0.212i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ 0.977 + 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7346457820 + 0.07880212555i\)
\(L(\frac12)\) \(\approx\) \(0.7346457820 + 0.07880212555i\)
\(L(1)\) \(\approx\) \(0.8537646622 + 0.07831252879i\)
\(L(1)\) \(\approx\) \(0.8537646622 + 0.07831252879i\)

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

−34.052879750643547875038204819427, −32.837009332734788122563231117469, −31.003175866229294384632274812934, −30.25520730955327809056404291128, −29.40691130836185517133865880968, −27.87533800315990927613267896293, −26.69928564567557605963665935002, −25.69599714181568494817211825990, −25.10749888907653808196508179081, −23.52693132969822039599588765439, −21.85394959777895625462113904678, −20.308752827342092615475207123080, −19.60706109536157083717021414396, −18.21447570072929577233856820812, −17.77114097409774969391322045793, −15.657392642777140282899853553069, −14.69828961158938254657900635226, −13.08615115697166879772288298220, −11.5294339791866932894092815712, −10.126304442709675692503652482150, −8.875847918728276374823036001518, −7.459525572024942080288397351283, −6.58271902165989171935951818355, −3.31242909604589351257165080516, −1.976892860955055929802562719617, 1.87477066186703555028755686561, 4.01937694936721663672995033725, 6.09580393980426574834670957512, 8.19289207499894776053676421584, 8.80243212592907273664511344483, 10.024974875402790407666301132341, 11.50602299006584212380386479379, 13.37728242222239024348209326304, 14.86513028836035638632047805480, 16.22368625890618663543349913677, 16.82115678715897378821291277663, 18.65824174946895463215534535692, 19.70818323076858278999327857732, 20.6912915274959466278915371478, 21.6211366176202009578040723279, 23.97202117047838411379662537577, 24.81738552530008975728125598051, 25.933500915403930600734360998571, 26.92148015160090313985594185967, 27.938403196128757102061269010780, 28.90257479253227193779614496080, 30.38381212526101821765200913863, 31.848126367508526217620097975761, 32.77813261768192280132165653089, 33.771893969647963234841978405620

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