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Properties

Label 1-209-209.112-r1-0-0
Degree $1$
Conductor $209$
Sign $0.914 + 0.404i$
Analytic cond. $22.4601$
Root an. cond. $22.4601$
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  + (−0.559 − 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (−0.241 + 0.970i)5-s + (−0.438 − 0.898i)6-s + (0.978 + 0.207i)7-s + (0.978 − 0.207i)8-s + (0.961 + 0.275i)9-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)12-s + (0.719 − 0.694i)13-s + (−0.374 − 0.927i)14-s + (−0.374 + 0.927i)15-s + (−0.719 − 0.694i)16-s + (−0.961 + 0.275i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (−0.241 + 0.970i)5-s + (−0.438 − 0.898i)6-s + (0.978 + 0.207i)7-s + (0.978 − 0.207i)8-s + (0.961 + 0.275i)9-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)12-s + (0.719 − 0.694i)13-s + (−0.374 − 0.927i)14-s + (−0.374 + 0.927i)15-s + (−0.719 − 0.694i)16-s + (−0.961 + 0.275i)17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ 0.914 + 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.102702610 + 0.4437743061i\)
\(L(\frac12)\) \(\approx\) \(2.102702610 + 0.4437743061i\)
\(L(1)\) \(\approx\) \(1.266306242 + 0.02085319644i\)
\(L(1)\) \(\approx\) \(1.266306242 + 0.02085319644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T \)
3 \( 1 + (0.990 + 0.139i)T \)
5 \( 1 + (-0.241 + 0.970i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (-0.961 + 0.275i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.615 + 0.788i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.0348 + 0.999i)T \)
53 \( 1 + (-0.241 - 0.970i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.997 + 0.0697i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (0.882 - 0.469i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.559 + 0.829i)T \)
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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

−26.693765707144586570633299418316, −25.25937639633539137863834041284, −24.718186464046558814524056968610, −23.97637603263955594790777521054, −23.20782415751240548106441780477, −21.39930650417274211273862817069, −20.54174544376795416866150290288, −19.7691584825403407229699475212, −18.76905956659381525941159900336, −17.830676465544178699376630474585, −16.80862552667162113182590050359, −15.79125784207204916039528511245, −15.02765256387713234520902003518, −13.89872604134805778660549701194, −13.30841209931861626327785857183, −11.726177612836877672474878141218, −10.38181622235460073300161595064, −8.97522485433367810145731019360, −8.62657434083877402315770587814, −7.67204642667235252042295355709, −6.573152924290193101590931176229, −4.91057412582576979566527431179, −4.16604296474035187244777401949, −1.97643012912373057090812045190, −0.88170165843980659836139651157, 1.41940752946402590672130285321, 2.61449187070130842242708719338, 3.49686156285539430418280607142, 4.69577830423631721338883999127, 6.84083360982859664504653098566, 8.02173304485057741459338336830, 8.55709130332801211506600350616, 9.86559703871512767267469418952, 10.81105171848619042753489580878, 11.546390758332044465646342136497, 13.01191392085680884170722209832, 13.88306430764779139516995609414, 14.97074142095757923753833386020, 15.76974210571537468221246793283, 17.46986313677962594258501723853, 18.22773175064586809042804790983, 19.011558729861845574308450834751, 19.92316962812446074157782626635, 20.76017848097125741974384957614, 21.586232879321117043415722846140, 22.42000258839715066804522059057, 23.73978004688510891211628615238, 25.12688992914478294649609366982, 25.76957601752935735244210765983, 26.85983177294324095708026282499

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