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Properties

Label 1-14e2-196.107-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: $\chi_{196} (107, \cdot )$
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

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

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