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Properties

Label 1-349-349.31-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.761 + 0.648i$
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  + (−0.947 + 0.319i)2-s + (−0.561 − 0.827i)3-s + (0.796 − 0.605i)4-s + (0.0541 − 0.998i)5-s + (0.796 + 0.605i)6-s + (0.976 + 0.214i)7-s + (−0.561 + 0.827i)8-s + (−0.370 + 0.928i)9-s + (0.267 + 0.963i)10-s + (−0.856 + 0.515i)11-s + (−0.947 − 0.319i)12-s + (−0.994 + 0.108i)13-s + (−0.994 + 0.108i)14-s + (−0.856 + 0.515i)15-s + (0.267 − 0.963i)16-s + (−0.561 − 0.827i)17-s + ⋯
L(s)  = 1  + (−0.947 + 0.319i)2-s + (−0.561 − 0.827i)3-s + (0.796 − 0.605i)4-s + (0.0541 − 0.998i)5-s + (0.796 + 0.605i)6-s + (0.976 + 0.214i)7-s + (−0.561 + 0.827i)8-s + (−0.370 + 0.928i)9-s + (0.267 + 0.963i)10-s + (−0.856 + 0.515i)11-s + (−0.947 − 0.319i)12-s + (−0.994 + 0.108i)13-s + (−0.994 + 0.108i)14-s + (−0.856 + 0.515i)15-s + (0.267 − 0.963i)16-s + (−0.561 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.761 + 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003243094748 + 0.008805322300i\)
\(L(\frac12)\) \(\approx\) \(0.003243094748 + 0.008805322300i\)
\(L(1)\) \(\approx\) \(0.4358200142 - 0.1023287701i\)
\(L(1)\) \(\approx\) \(0.4358200142 - 0.1023287701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.947 + 0.319i)T \)
3 \( 1 + (-0.561 - 0.827i)T \)
5 \( 1 + (0.0541 - 0.998i)T \)
7 \( 1 + (0.976 + 0.214i)T \)
11 \( 1 + (-0.856 + 0.515i)T \)
13 \( 1 + (-0.994 + 0.108i)T \)
17 \( 1 + (-0.561 - 0.827i)T \)
19 \( 1 + (-0.161 + 0.986i)T \)
23 \( 1 + (-0.994 + 0.108i)T \)
29 \( 1 + (-0.947 + 0.319i)T \)
31 \( 1 + (0.907 - 0.419i)T \)
37 \( 1 + (-0.994 + 0.108i)T \)
41 \( 1 + (-0.561 + 0.827i)T \)
43 \( 1 + (-0.856 + 0.515i)T \)
47 \( 1 + (-0.725 - 0.687i)T \)
53 \( 1 + (0.468 - 0.883i)T \)
59 \( 1 + (-0.561 - 0.827i)T \)
61 \( 1 + (-0.947 - 0.319i)T \)
67 \( 1 + (-0.161 - 0.986i)T \)
71 \( 1 + (-0.561 + 0.827i)T \)
73 \( 1 + (0.976 + 0.214i)T \)
79 \( 1 + (0.468 + 0.883i)T \)
83 \( 1 + (0.0541 - 0.998i)T \)
89 \( 1 + (0.647 + 0.762i)T \)
97 \( 1 + (0.0541 + 0.998i)T \)
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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

−26.008912825059930911062391613195, −24.42205017584801119954345836415, −23.75860366845156086170551481167, −22.37233981781870830777684166251, −21.70311209348055773560435662973, −21.08693260163617600226896591488, −20.06710428372727124602517753466, −19.10004219511777687193067413211, −17.98959697101382891026519887239, −17.568729694460278076070934746961, −16.72081795191039139738781862582, −15.431263859016550850847898343243, −15.1007332174426009326602695628, −13.74835650106664619685493694972, −12.173657482592982786956571124, −11.32751228980520643159498199169, −10.59177940008432310195320209320, −10.16326350393429785805173560461, −8.873749697689942013351657450, −7.85041425104591937766470627259, −6.84653262372954306270143018115, −5.70097426809851552846147567212, −4.381903771743354486895444680156, −3.13389120637932691088588154842, −2.0434170004891960594841527174, 0.00788165180593645566537920261, 1.600995691658567275988764816073, 2.237853400552602278129417874931, 4.886480815455946648823584950946, 5.35994304297070333548428769952, 6.6479061087940307617483278757, 7.880041519924248036231829863817, 8.08932809350821310188511845316, 9.45796260743734584711858701450, 10.44461660982280595009835407946, 11.69696844491806420688913549278, 12.11598297625726939474929943790, 13.33985685629776943806790017343, 14.48375020182816849358670159231, 15.60071765662860900941974888334, 16.58711057539447281872052562326, 17.25802581689719734818870501928, 18.03377897265967945935266063344, 18.6218648040945618859026504180, 19.840733030404054720712016885153, 20.47004466037394022465384079708, 21.441589524612490177133603433074, 22.898780697583815255258142551, 23.82649389160525695702008863577, 24.53292944131889560903566581940

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