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Properties

Label 1-14e2-196.11-r1-0-0
Degree $1$
Conductor $196$
Sign $0.0320 + 0.999i$
Analytic cond. $21.0631$
Root an. cond. $21.0631$
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  + (−0.955 + 0.294i)3-s + (−0.733 − 0.680i)5-s + (0.826 − 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.900 + 0.433i)15-s + (0.365 − 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.623 + 0.781i)27-s + (0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (0.955 + 0.294i)33-s + (−0.988 + 0.149i)37-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)3-s + (−0.733 − 0.680i)5-s + (0.826 − 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.900 + 0.433i)15-s + (0.365 − 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.623 + 0.781i)27-s + (0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (0.955 + 0.294i)33-s + (−0.988 + 0.149i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.0320 + 0.999i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ 0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3125636640 + 0.3027010284i\)
\(L(\frac12)\) \(\approx\) \(0.3125636640 + 0.3027010284i\)
\(L(1)\) \(\approx\) \(0.5869937518 + 0.01380845686i\)
\(L(1)\) \(\approx\) \(0.5869937518 + 0.01380845686i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.955 + 0.294i)T \)
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

−26.68819158718837010900289024282, −25.60518497811778623031886755103, −24.35833269427178776723829162935, −23.54160507786991207261756796681, −22.79848520402593273092240072775, −22.07505006540258244159859052083, −20.90593810381375122403251071153, −19.54644354709641519114444389828, −18.79741387415668086196981945570, −17.83724915375833229726616864208, −17.01408832538336471551178259027, −15.746470929003785621664260094128, −15.11045399081632505466822988042, −13.71133832432518759525711131224, −12.37912329023433015279738276473, −11.89216828131520462671281944392, −10.56558563570810534014318344890, −10.02833097869441829046197584645, −7.91011936929446369467415733867, −7.38828614025601610841084152793, −6.09337827363332894458681957169, −5.01324350532338132201914963149, −3.68598829258360169828607708573, −2.085778007766857620117333975882, −0.225056223073504245348415084917, 0.91371576073018344554973944594, 3.046929713269906925790319708504, 4.674731583914145497162423174912, 5.116517243485297009225965083708, 6.65949759333118529032318133201, 7.751623102115310801041767263921, 9.04777607292324568020756141900, 10.18761841641411726534858210953, 11.31935356153211228881183808198, 12.08134676880889691179560264756, 12.96514536608638030693500559236, 14.38155698274451409595688565277, 15.90364656400539342638908934352, 16.08345492020957117787932501504, 17.22492433400843142127755095936, 18.24899203147055019195610852126, 19.28201953448851896648664564585, 20.42241321531214173813653751470, 21.3396168272033517541107595437, 22.29017994052854514729973110676, 23.28481500640571686625692314328, 24.02397859378360072276453925640, 24.72082349342048516341248778506, 26.43662660399918032840880893348, 27.01928218658977626566419321860

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