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Properties

Label 1-3272-3272.1139-r0-0-0
Degree $1$
Conductor $3272$
Sign $-0.723 + 0.690i$
Analytic cond. $15.1951$
Root an. cond. $15.1951$
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  + (−0.361 − 0.932i)3-s + (0.850 + 0.526i)5-s + (−0.707 + 0.707i)7-s + (−0.739 + 0.673i)9-s + (0.998 + 0.0461i)11-s + (0.0461 + 0.998i)13-s + (0.183 − 0.982i)15-s + (−0.982 − 0.183i)17-s + (0.948 − 0.317i)19-s + (0.914 + 0.403i)21-s + (−0.183 + 0.982i)23-s + (0.445 + 0.895i)25-s + (0.895 + 0.445i)27-s + (−0.769 + 0.638i)29-s + (−0.707 + 0.707i)31-s + ⋯
L(s)  = 1  + (−0.361 − 0.932i)3-s + (0.850 + 0.526i)5-s + (−0.707 + 0.707i)7-s + (−0.739 + 0.673i)9-s + (0.998 + 0.0461i)11-s + (0.0461 + 0.998i)13-s + (0.183 − 0.982i)15-s + (−0.982 − 0.183i)17-s + (0.948 − 0.317i)19-s + (0.914 + 0.403i)21-s + (−0.183 + 0.982i)23-s + (0.445 + 0.895i)25-s + (0.895 + 0.445i)27-s + (−0.769 + 0.638i)29-s + (−0.707 + 0.707i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3272 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3272 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3272\)    =    \(2^{3} \cdot 409\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(15.1951\)
Root analytic conductor: \(15.1951\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3272} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3272,\ (0:\ ),\ -0.723 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2809046853 + 0.7007593962i\)
\(L(\frac12)\) \(\approx\) \(0.2809046853 + 0.7007593962i\)
\(L(1)\) \(\approx\) \(0.8824912694 + 0.08547272704i\)
\(L(1)\) \(\approx\) \(0.8824912694 + 0.08547272704i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
409 \( 1 \)
good3 \( 1 + (-0.361 - 0.932i)T \)
5 \( 1 + (0.850 + 0.526i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.998 + 0.0461i)T \)
13 \( 1 + (0.0461 + 0.998i)T \)
17 \( 1 + (-0.982 - 0.183i)T \)
19 \( 1 + (0.948 - 0.317i)T \)
23 \( 1 + (-0.183 + 0.982i)T \)
29 \( 1 + (-0.769 + 0.638i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.873 + 0.486i)T \)
41 \( 1 + (0.995 - 0.0922i)T \)
43 \( 1 + (-0.948 - 0.317i)T \)
47 \( 1 + (-0.873 + 0.486i)T \)
53 \( 1 - T \)
59 \( 1 + (0.990 - 0.138i)T \)
61 \( 1 + (-0.914 + 0.403i)T \)
67 \( 1 + (-0.0461 + 0.998i)T \)
71 \( 1 + (0.850 - 0.526i)T \)
73 \( 1 + (0.138 - 0.990i)T \)
79 \( 1 + (-0.228 - 0.973i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (0.228 + 0.973i)T \)
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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

−18.400170561771318160541562449732, −17.72248922725968029476866997404, −17.06151419293677496964036833208, −16.66059205865833776870807753414, −16.00291717204945287519599022320, −15.27024324787211722484388781633, −14.39977958742726574347883865452, −13.84732653159818989302243763959, −12.95305836559274279223278807437, −12.48319731979074014220693780879, −11.40633480849301747736567106782, −10.818436633237464077048931906666, −9.94063792917304795810344960335, −9.63851257503316620961411326377, −8.92752692721671939205776457534, −8.10618046755662134641855196780, −6.91296691595979421138435007698, −6.22368399739076954539099181247, −5.647338965723756705586924606953, −4.81122066938028358524342395052, −4.006408232744158912095267949850, −3.42231091187303642364088942153, −2.37571337453983947490322874718, −1.19589615673458191317287956895, −0.2308775418301570699518476867, 1.5077944588919337240843022194, 1.83035871579395447452253050815, 2.85385330133245599415356503166, 3.554259493941592968714556660278, 4.90906814362139434527038914709, 5.64106397388450572243906691420, 6.35325881347632210255010228839, 6.85145658024617916205021025222, 7.36124148475493204371720511164, 8.7366424510474081216945395718, 9.1940605383755623955082111212, 9.7970240495604529366063124684, 10.961575007160487838678859478654, 11.5031913585493476154444548247, 12.10523157108291784480313770209, 12.96197677207556420746669188840, 13.54216850122196055935432867337, 14.1382388268031632958955839980, 14.77991001309024779717961673398, 15.82811841085034561524548089055, 16.47926249925280402730738081335, 17.2487363782339160985441142255, 17.89244074486194503348090961832, 18.337617391769906774370930381505, 19.12815024453346642198885266110

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