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Properties

Label 1-349-349.346-r0-0-0
Degree $1$
Conductor $349$
Sign $0.318 - 0.948i$
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  + (−0.891 − 0.452i)2-s + (0.935 − 0.353i)3-s + (0.590 + 0.806i)4-s + (−0.968 + 0.250i)5-s + (−0.994 − 0.108i)6-s + (−0.999 − 0.0361i)7-s + (−0.161 − 0.986i)8-s + (0.750 − 0.661i)9-s + (0.976 + 0.214i)10-s + (0.907 + 0.419i)11-s + (0.837 + 0.546i)12-s + (−0.0180 − 0.999i)13-s + (0.874 + 0.484i)14-s + (−0.817 + 0.576i)15-s + (−0.302 + 0.953i)16-s + (−0.161 + 0.986i)17-s + ⋯
L(s)  = 1  + (−0.891 − 0.452i)2-s + (0.935 − 0.353i)3-s + (0.590 + 0.806i)4-s + (−0.968 + 0.250i)5-s + (−0.994 − 0.108i)6-s + (−0.999 − 0.0361i)7-s + (−0.161 − 0.986i)8-s + (0.750 − 0.661i)9-s + (0.976 + 0.214i)10-s + (0.907 + 0.419i)11-s + (0.837 + 0.546i)12-s + (−0.0180 − 0.999i)13-s + (0.874 + 0.484i)14-s + (−0.817 + 0.576i)15-s + (−0.302 + 0.953i)16-s + (−0.161 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.318 - 0.948i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.318 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7404921708 - 0.5325382673i\)
\(L(\frac12)\) \(\approx\) \(0.7404921708 - 0.5325382673i\)
\(L(1)\) \(\approx\) \(0.7744951065 - 0.2735363169i\)
\(L(1)\) \(\approx\) \(0.7744951065 - 0.2735363169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.891 - 0.452i)T \)
3 \( 1 + (0.935 - 0.353i)T \)
5 \( 1 + (-0.968 + 0.250i)T \)
7 \( 1 + (-0.999 - 0.0361i)T \)
11 \( 1 + (0.907 + 0.419i)T \)
13 \( 1 + (-0.0180 - 0.999i)T \)
17 \( 1 + (-0.161 + 0.986i)T \)
19 \( 1 + (0.958 + 0.284i)T \)
23 \( 1 + (-0.0180 - 0.999i)T \)
29 \( 1 + (0.837 - 0.546i)T \)
31 \( 1 + (-0.561 - 0.827i)T \)
37 \( 1 + (-0.856 + 0.515i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (-0.0901 - 0.995i)T \)
47 \( 1 + (0.796 + 0.605i)T \)
53 \( 1 + (0.647 + 0.762i)T \)
59 \( 1 + (0.935 - 0.353i)T \)
61 \( 1 + (0.0541 - 0.998i)T \)
67 \( 1 + (-0.725 - 0.687i)T \)
71 \( 1 + (0.935 + 0.353i)T \)
73 \( 1 + (0.530 - 0.847i)T \)
79 \( 1 + (0.647 - 0.762i)T \)
83 \( 1 + (0.700 + 0.713i)T \)
89 \( 1 + (-0.619 + 0.785i)T \)
97 \( 1 + (-0.968 - 0.250i)T \)
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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

−25.128055754751630610470317049229, −24.45050336087768870801331887511, −23.51087114640428569545773568505, −22.4353359727185534825074197685, −21.29722044717590514970064158578, −20.04473605027057452466623223238, −19.66258709937104848220461692950, −19.08878265874978947198238613680, −18.08321143105084711117899228141, −16.43923578064224589399675725844, −16.249449397492959137788545390380, −15.43615055761141275261544986868, −14.38182043258172509621394583837, −13.5976093155238620385327428193, −12.05089386823257964602390758714, −11.233045166499916146811681627078, −9.883204603264881580405330484049, −9.14966870348178143735673377539, −8.61007548573916644071464738543, −7.29094782673448851242145251377, −6.828152547623630823315037238854, −5.15396533382025095748362418929, −3.82327100335898272018601453412, −2.86522384010745854845814083087, −1.22737469056727396184158256956, 0.80255043491645684081822056352, 2.33670911546732574643497701692, 3.407294106530383938974819954770, 3.96084333876213370657139477307, 6.4026148662366086308621577531, 7.20191094574269001576826379653, 8.06591274581888802020391621898, 8.88428252665898103988261454253, 9.86785254813649915823241787553, 10.689172437215092416045136387578, 12.2669477592735049877125415827, 12.38337088445178550677440231884, 13.70861677456091856347083494567, 15.090383265417041473244877882613, 15.59948905174891513034061581549, 16.686723534937139332741196181579, 17.77921371479088361837861941395, 18.86340438054275891765708080096, 19.27086644187730137435618422734, 20.16801845073442899345839541165, 20.468211290170886607437107630338, 22.073725648116904872167454078651, 22.68528039568225407261512866661, 24.06372772291422625514505706689, 24.96924492380093697461869955552

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