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Properties

Label 1-337-337.192-r0-0-0
Degree $1$
Conductor $337$
Sign $0.681 + 0.732i$
Analytic cond. $1.56502$
Root an. cond. $1.56502$
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  + (−0.733 + 0.680i)2-s + (0.997 + 0.0747i)3-s + (0.0747 − 0.997i)4-s + (0.993 − 0.111i)5-s + (−0.781 + 0.623i)6-s + (−0.974 + 0.222i)7-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)9-s + (−0.652 + 0.757i)10-s + (0.846 + 0.532i)11-s + (0.149 − 0.988i)12-s + (0.365 − 0.930i)13-s + (0.563 − 0.826i)14-s + (0.999 − 0.0373i)15-s + (−0.988 − 0.149i)16-s + (−0.111 + 0.993i)17-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (0.997 + 0.0747i)3-s + (0.0747 − 0.997i)4-s + (0.993 − 0.111i)5-s + (−0.781 + 0.623i)6-s + (−0.974 + 0.222i)7-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)9-s + (−0.652 + 0.757i)10-s + (0.846 + 0.532i)11-s + (0.149 − 0.988i)12-s + (0.365 − 0.930i)13-s + (0.563 − 0.826i)14-s + (0.999 − 0.0373i)15-s + (−0.988 − 0.149i)16-s + (−0.111 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(337\)
Sign: $0.681 + 0.732i$
Analytic conductor: \(1.56502\)
Root analytic conductor: \(1.56502\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{337} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 337,\ (0:\ ),\ 0.681 + 0.732i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319849804 + 0.5747838064i\)
\(L(\frac12)\) \(\approx\) \(1.319849804 + 0.5747838064i\)
\(L(1)\) \(\approx\) \(1.123987432 + 0.3458344877i\)
\(L(1)\) \(\approx\) \(1.123987432 + 0.3458344877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad337 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
3 \( 1 + (0.997 + 0.0747i)T \)
5 \( 1 + (0.993 - 0.111i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (0.846 + 0.532i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.111 + 0.993i)T \)
19 \( 1 + (-0.884 - 0.467i)T \)
23 \( 1 + (0.593 + 0.804i)T \)
29 \( 1 + (-0.399 + 0.916i)T \)
31 \( 1 + (-0.467 - 0.884i)T \)
37 \( 1 + (-0.0747 - 0.997i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (0.399 + 0.916i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.884 - 0.467i)T \)
67 \( 1 + (-0.804 + 0.593i)T \)
71 \( 1 + (-0.884 + 0.467i)T \)
73 \( 1 + (0.0373 + 0.999i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.185 + 0.982i)T \)
89 \( 1 + (0.399 - 0.916i)T \)
97 \( 1 + (-0.993 - 0.111i)T \)
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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

−25.32088926616399310922948398970, −24.3845734362073991361241464660, −22.73871693450541943346570070112, −21.88027242448333349905979567313, −21.05778307111016725828473689908, −20.437794748280860597591583448771, −19.261636571405327203240725383513, −18.96042930605276528293308134995, −17.960170649878662741681111025508, −16.71527528791224403998911900776, −16.258363288314933685111059059312, −14.69257082215136535210184772421, −13.67475995916060174543580900034, −13.17563152373272303949069218666, −12.07309918133319073303527350074, −10.77299761368205917475893881845, −9.80171677759070565346387237337, −9.19571678885120714001063358244, −8.525307596553618218269657765271, −7.00068824869245471197883179865, −6.425154546901828431466232297050, −4.29962793891165663355701605688, −3.27389427892967009858511102762, −2.3652865157974639780536729168, −1.26292821584271450578319987143, 1.42191657880227430859931490320, 2.4649656171714407093444052559, 3.896053808308504160736768376423, 5.45630376874064800300411098109, 6.42281968801079472099381908118, 7.27299936513783781557993147132, 8.60758103824532193730160534640, 9.19322748125562622365212486277, 9.92457179894312502064133228554, 10.77774484072424062868939026944, 12.81979930993012349013993021338, 13.27688558505801771266643147141, 14.56634551104429591799904910786, 15.06832794413400646297693243596, 16.07058002127199103036284143158, 17.08756285803009194077837791131, 17.84046508565825204966265021670, 18.89303886195363163820571114683, 19.6529080317789600633245294974, 20.30827979515047741470412384055, 21.498692273394338498917567959166, 22.40063255076939271885397829833, 23.550734392236613830980070433011, 24.75643238261493262509653789573, 25.23507084994068201663612611681

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