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Properties

Label 1-349-349.254-r0-0-0
Degree $1$
Conductor $349$
Sign $0.751 - 0.659i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.935 − 0.353i)2-s + (−0.999 − 0.0361i)3-s + (0.750 − 0.661i)4-s + (0.958 − 0.284i)5-s + (−0.947 + 0.319i)6-s + (0.590 + 0.806i)7-s + (0.468 − 0.883i)8-s + (0.997 + 0.0721i)9-s + (0.796 − 0.605i)10-s + (0.267 − 0.963i)11-s + (−0.773 + 0.633i)12-s + (−0.891 + 0.452i)13-s + (0.837 + 0.546i)14-s + (−0.968 + 0.250i)15-s + (0.126 − 0.992i)16-s + (0.468 + 0.883i)17-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + (−0.999 − 0.0361i)3-s + (0.750 − 0.661i)4-s + (0.958 − 0.284i)5-s + (−0.947 + 0.319i)6-s + (0.590 + 0.806i)7-s + (0.468 − 0.883i)8-s + (0.997 + 0.0721i)9-s + (0.796 − 0.605i)10-s + (0.267 − 0.963i)11-s + (−0.773 + 0.633i)12-s + (−0.891 + 0.452i)13-s + (0.837 + 0.546i)14-s + (−0.968 + 0.250i)15-s + (0.126 − 0.992i)16-s + (0.468 + 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.751 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.975927380 - 0.7444103245i\)
\(L(\frac12)\) \(\approx\) \(1.975927380 - 0.7444103245i\)
\(L(1)\) \(\approx\) \(1.618291116 - 0.4208572669i\)
\(L(1)\) \(\approx\) \(1.618291116 - 0.4208572669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.935 - 0.353i)T \)
3 \( 1 + (-0.999 - 0.0361i)T \)
5 \( 1 + (0.958 - 0.284i)T \)
7 \( 1 + (0.590 + 0.806i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (-0.891 + 0.452i)T \)
17 \( 1 + (0.468 + 0.883i)T \)
19 \( 1 + (0.336 + 0.941i)T \)
23 \( 1 + (-0.891 + 0.452i)T \)
29 \( 1 + (-0.773 - 0.633i)T \)
31 \( 1 + (0.976 + 0.214i)T \)
37 \( 1 + (0.0541 - 0.998i)T \)
41 \( 1 + (0.468 - 0.883i)T \)
43 \( 1 + (0.700 + 0.713i)T \)
47 \( 1 + (-0.370 - 0.928i)T \)
53 \( 1 + (-0.856 - 0.515i)T \)
59 \( 1 + (-0.999 - 0.0361i)T \)
61 \( 1 + (-0.161 - 0.986i)T \)
67 \( 1 + (0.647 + 0.762i)T \)
71 \( 1 + (-0.999 + 0.0361i)T \)
73 \( 1 + (0.403 - 0.915i)T \)
79 \( 1 + (-0.856 + 0.515i)T \)
83 \( 1 + (-0.232 + 0.972i)T \)
89 \( 1 + (-0.0901 + 0.995i)T \)
97 \( 1 + (0.958 + 0.284i)T \)
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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

−24.57522739318766603239457195140, −24.10347646082447263565418670283, −22.95348972770435602659403452011, −22.45207184088623430649622878503, −21.73517688608064428372397447894, −20.745555092371878693102100613, −20.07968718546694377531055008448, −18.30730841993409034765757098213, −17.39099831762406919765291201643, −17.13425825309913509085988767090, −16.00238191014704300175514102911, −14.90524190592538069455353806015, −14.12586035392415561435705421708, −13.20890551243477406764417715964, −12.29704459240804491250420431211, −11.41593619109575608282951602614, −10.437790749212156451877748185253, −9.62771394537232798006501045370, −7.59635663429424726465713522308, −7.02993559023578708668459965641, −6.03606711208107763562039373333, −4.956273425151800467865422564, −4.492511525218399934974927117371, −2.793665764909179286846576232796, −1.49545165303549240898243296196, 1.382189189822857105722021249453, 2.23263337498643792284416305571, 3.88443889395475495657563505612, 5.06457718246371742996108605347, 5.78593137626546064141675507234, 6.28080024898363471077646765024, 7.81145400301388348472232491936, 9.42017489608840969233934235849, 10.26898167823672558147061821367, 11.29291777613898358922384899909, 12.094893911775709347209379409200, 12.71355750859246165347010952218, 13.90918115886586262882196953138, 14.57615574815463006779004650792, 15.80132156200586970785204806724, 16.67760623137797884820981259744, 17.49652023849197748384615266790, 18.60431971610491137622454000531, 19.354970565752533142883390078524, 20.880054500855743348595226020673, 21.476457659729703782350531567928, 21.92253737296938096622766187249, 22.77672858281375987086072075325, 24.00443107984745150621564144635, 24.44151948725457647028166019274

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