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Properties

Label 1-17-17.2-r0-0-0
Degree $1$
Conductor $17$
Sign $0.739 + 0.673i$
Analytic cond. $0.0789476$
Root an. cond. $0.0789476$
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  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − i·8-s − i·9-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + (0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯
L(s)  = 1  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − i·8-s − i·9-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + (0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(0.0789476\)
Root analytic conductor: \(0.0789476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (0:\ ),\ 0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5807245684 + 0.2248635705i\)
\(L(\frac12)\) \(\approx\) \(0.5807245684 + 0.2248635705i\)
\(L(1)\) \(\approx\) \(0.8412819557 + 0.2607920837i\)
\(L(1)\) \(\approx\) \(0.8412819557 + 0.2607920837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 - T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−41.223565741636124516117175386745, −39.54038332050637170033840409929, −38.743939439628778238940232207165, −37.59076506490882900091239033580, −36.35268278308999465632675897784, −34.97795633295879432431146639064, −32.344940951259973472144765426760, −31.95109387779334854941971199316, −30.61290869301841514582147870720, −28.77816803021989973591876294860, −27.611458380906137526811615674624, −26.54501481103078000322250516549, −24.62558193307131620423413490107, −22.50091907098631848647446969433, −21.38249265328479352279990162659, −19.862604940973289620007646282, −19.17269032844112991056189821387, −16.65682053984199700949071927411, −14.96366013445572926040451475996, −13.11541261636260959551172937378, −11.57473470221158547204510748840, −9.58696179846013643547410728982, −8.533109263551343292679234102286, −4.71512875016980910626184728491, −3.03085021754411121651803365475, 3.73234531397038605182942743202, 6.761948321041484672115641845354, 7.64497211191416669983437925264, 9.63398131799367961856970535390, 12.50111717073719875221216019938, 14.1405308258296735664314752950, 15.199466723475587911399842111731, 17.095396259645679961870329884163, 18.759036754209906446697417528529, 19.83031510123342842902457411670, 22.5283052827079345732345438771, 23.55435493382188059643245697127, 25.03639525135445337989798997153, 26.18826393503240730471673423900, 27.23645078646237448660705955298, 29.676538417633617230501409897540, 31.00830660795214613583261499309, 32.094954838183972985893136613982, 33.64124875849802388347511618307, 35.23177510386508830173456058388, 35.856980322047274678357208150409, 37.34202697322947125296670875412, 39.05857314286053238935053734852, 40.98296614454602221763597177948, 42.061728316424146917429279822682

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