↑

Properties

Label 1-349-349.19-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.122 + 0.992i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.968 − 0.250i)2-s + (0.958 + 0.284i)3-s + (0.874 + 0.484i)4-s + (−0.674 − 0.738i)5-s + (−0.856 − 0.515i)6-s + (0.336 + 0.941i)7-s + (−0.725 − 0.687i)8-s + (0.837 + 0.546i)9-s + (0.468 + 0.883i)10-s + (−0.561 + 0.827i)11-s + (0.700 + 0.713i)12-s + (−0.817 + 0.576i)13-s + (−0.0901 − 0.995i)14-s + (−0.436 − 0.899i)15-s + (0.530 + 0.847i)16-s + (−0.725 + 0.687i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.250i)2-s + (0.958 + 0.284i)3-s + (0.874 + 0.484i)4-s + (−0.674 − 0.738i)5-s + (−0.856 − 0.515i)6-s + (0.336 + 0.941i)7-s + (−0.725 − 0.687i)8-s + (0.837 + 0.546i)9-s + (0.468 + 0.883i)10-s + (−0.561 + 0.827i)11-s + (0.700 + 0.713i)12-s + (−0.817 + 0.576i)13-s + (−0.0901 − 0.995i)14-s + (−0.436 − 0.899i)15-s + (0.530 + 0.847i)16-s + (−0.725 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4890377517 + 0.5530573633i\)
\(L(\frac12)\) \(\approx\) \(0.4890377517 + 0.5530573633i\)
\(L(1)\) \(\approx\) \(0.7361274663 + 0.1751318106i\)
\(L(1)\) \(\approx\) \(0.7361274663 + 0.1751318106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.968 - 0.250i)T \)
3 \( 1 + (0.958 + 0.284i)T \)
5 \( 1 + (-0.674 - 0.738i)T \)
7 \( 1 + (0.336 + 0.941i)T \)
11 \( 1 + (-0.561 + 0.827i)T \)
13 \( 1 + (-0.817 + 0.576i)T \)
17 \( 1 + (-0.725 + 0.687i)T \)
19 \( 1 + (-0.922 - 0.386i)T \)
23 \( 1 + (-0.817 + 0.576i)T \)
29 \( 1 + (0.700 - 0.713i)T \)
31 \( 1 + (-0.161 + 0.986i)T \)
37 \( 1 + (0.907 + 0.419i)T \)
41 \( 1 + (-0.725 - 0.687i)T \)
43 \( 1 + (0.997 + 0.0721i)T \)
47 \( 1 + (-0.994 - 0.108i)T \)
53 \( 1 + (-0.370 - 0.928i)T \)
59 \( 1 + (0.958 + 0.284i)T \)
61 \( 1 + (0.267 - 0.963i)T \)
67 \( 1 + (0.796 + 0.605i)T \)
71 \( 1 + (0.958 - 0.284i)T \)
73 \( 1 + (-0.983 - 0.179i)T \)
79 \( 1 + (-0.370 + 0.928i)T \)
83 \( 1 + (-0.302 + 0.953i)T \)
89 \( 1 + (0.750 + 0.661i)T \)
97 \( 1 + (-0.674 + 0.738i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.63894185231216237458382828957, −24.00487622616311688796491472891, −23.218763273956495800589664463117, −21.79008209914867078100397480762, −20.588086150713801407012686482261, −19.97824219565789577736133403939, −19.28833359733310457312510911813, −18.445179850069275770455460441446, −17.76213648223368354748586416716, −16.49657138590253975860114912534, −15.6544037593840040322530214697, −14.711761763874937631880255288887, −14.14923764830265554938374608372, −12.87260294510149710638922245665, −11.53395720342868947549694298824, −10.60919838363503145979706502293, −9.92816147518922796226981489450, −8.55090851826305761801799275070, −7.879439844547796507363214007251, −7.24641881097589728631923206996, −6.28117411593813616186871190993, −4.42257007042087478255578311347, −3.09852371497946222262662232894, −2.230618058667638153460916521282, −0.520840891989253822458716070865, 1.829089823979637771042958055706, 2.47571040520696357613247150841, 3.96966930697178096792171443198, 4.95109903615996682014295198953, 6.76272794577535493536594207856, 7.910920662103667171201368519185, 8.43867264717229282797156900640, 9.27129166878799422165267312873, 10.09286666326718047180172000313, 11.34627531948458865170386524242, 12.328194739277680632238856937737, 12.98360757240082084014754014609, 14.658473550644188653377734418578, 15.462927143007207301589683155265, 15.91393774651422557510672118581, 17.16859930262559717629764873788, 18.049303819219087250457755855063, 19.25382564219646278554928315894, 19.56354427924738491796616148576, 20.497832689641368142015177196670, 21.29440427915157452554974210390, 21.92959653124392078335758795777, 23.77128915217333121642206083277, 24.41241098909466997407399625419, 25.2935154114210969553837507937

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