↑

Properties

Label 1-349-349.38-r1-0-0
Degree $1$
Conductor $349$
Sign $-0.895 - 0.444i$
Analytic cond. $37.5052$
Root an. cond. $37.5052$
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  + (−0.963 − 0.267i)2-s + (0.725 + 0.687i)3-s + (0.856 + 0.515i)4-s + (−0.976 − 0.214i)5-s + (−0.515 − 0.856i)6-s + (−0.762 + 0.647i)7-s + (−0.687 − 0.725i)8-s + (0.0541 + 0.998i)9-s + (0.883 + 0.468i)10-s + (−0.827 + 0.561i)11-s + (0.267 + 0.963i)12-s + (0.419 + 0.907i)13-s + (0.907 − 0.419i)14-s + (−0.561 − 0.827i)15-s + (0.468 + 0.883i)16-s + (0.725 + 0.687i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.267i)2-s + (0.725 + 0.687i)3-s + (0.856 + 0.515i)4-s + (−0.976 − 0.214i)5-s + (−0.515 − 0.856i)6-s + (−0.762 + 0.647i)7-s + (−0.687 − 0.725i)8-s + (0.0541 + 0.998i)9-s + (0.883 + 0.468i)10-s + (−0.827 + 0.561i)11-s + (0.267 + 0.963i)12-s + (0.419 + 0.907i)13-s + (0.907 − 0.419i)14-s + (−0.561 − 0.827i)15-s + (0.468 + 0.883i)16-s + (0.725 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(37.5052\)
Root analytic conductor: \(37.5052\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (1:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1186669830 + 0.5060168909i\)
\(L(\frac12)\) \(\approx\) \(-0.1186669830 + 0.5060168909i\)
\(L(1)\) \(\approx\) \(0.5663269648 + 0.2705567661i\)
\(L(1)\) \(\approx\) \(0.5663269648 + 0.2705567661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.963 - 0.267i)T \)
3 \( 1 + (0.725 + 0.687i)T \)
5 \( 1 + (-0.976 - 0.214i)T \)
7 \( 1 + (-0.762 + 0.647i)T \)
11 \( 1 + (-0.827 + 0.561i)T \)
13 \( 1 + (0.419 + 0.907i)T \)
17 \( 1 + (0.725 + 0.687i)T \)
19 \( 1 + (0.796 + 0.605i)T \)
23 \( 1 + (0.907 - 0.419i)T \)
29 \( 1 + (-0.267 + 0.963i)T \)
31 \( 1 + (-0.161 - 0.986i)T \)
37 \( 1 + (-0.907 + 0.419i)T \)
41 \( 1 + (-0.725 + 0.687i)T \)
43 \( 1 + (-0.827 + 0.561i)T \)
47 \( 1 + (-0.108 - 0.994i)T \)
53 \( 1 + (-0.928 - 0.370i)T \)
59 \( 1 + (0.687 - 0.725i)T \)
61 \( 1 + (-0.963 + 0.267i)T \)
67 \( 1 + (0.796 - 0.605i)T \)
71 \( 1 + (0.687 + 0.725i)T \)
73 \( 1 + (-0.647 - 0.762i)T \)
79 \( 1 + (0.928 - 0.370i)T \)
83 \( 1 + (-0.976 - 0.214i)T \)
89 \( 1 + (-0.319 + 0.947i)T \)
97 \( 1 + (0.214 + 0.976i)T \)
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

−24.214006921694001417221956240821, −23.456632764083874940380327235145, −22.80802756496768878495516228713, −20.91718330257977446653773020997, −20.25873509529445185331377128676, −19.48899256661848422531528466141, −18.854546360661976722481004356216, −18.168215603517093558905696743002, −17.05343648899665600163477395484, −15.74188370430502684567016003332, −15.61028971461080722654146675651, −14.23800338070936226532085427451, −13.2984837451689719300499469630, −12.21570470291020729777289093892, −11.14011973521246123254193898425, −10.205838524308280325183163257183, −9.11524048419271513212589582693, −8.12567988905258148273181572678, −7.447937562096065755660606074871, −6.81416832851414704836066439431, −5.446101466490343925661219840994, −3.34817239625888996316289591844, −2.92578415322811841840628990227, −1.02266788222436461776371617842, −0.21932062203298794063449887662, 1.702714770244701283141575622023, 3.04774580577665518664839240306, 3.688747243344531801819266985413, 5.1437366707056494334332852408, 6.77283107879239276388785296929, 7.847953048711385318445915136086, 8.55165603924255425521351015164, 9.43462714880107709930611155060, 10.22150157894539270630582755767, 11.26515836122372100722530051273, 12.26495591640537367880293717527, 13.12441306049090135601038267375, 14.7947647960820986963178161267, 15.4378497586976077412070480469, 16.27213927093896198875391371606, 16.730372866900820736706014860722, 18.58980420708979924159188393337, 18.81719692728793329147722737251, 19.8023384064244035569993934759, 20.5589297007605703301181899541, 21.24435720191478650467188580800, 22.269325289597051754809300110820, 23.40327353304849535405710188257, 24.55574105209636490925957328770, 25.49065596512790779765518066983

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