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Properties

Label 1-103-103.80-r1-0-0
Degree $1$
Conductor $103$
Sign $-0.494 - 0.869i$
Analytic cond. $11.0688$
Root an. cond. $11.0688$
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  + (−0.602 + 0.798i)2-s + (−0.445 − 0.895i)3-s + (−0.273 − 0.961i)4-s + (−0.0922 − 0.995i)5-s + (0.982 + 0.183i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (0.850 + 0.526i)10-s + (0.602 − 0.798i)11-s + (−0.739 + 0.673i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.850 + 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (−0.445 − 0.895i)3-s + (−0.273 − 0.961i)4-s + (−0.0922 − 0.995i)5-s + (0.982 + 0.183i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (0.850 + 0.526i)10-s + (0.602 − 0.798i)11-s + (−0.739 + 0.673i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.850 + 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.494 - 0.869i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ -0.494 - 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4716602858 - 0.8105820363i\)
\(L(\frac12)\) \(\approx\) \(0.4716602858 - 0.8105820363i\)
\(L(1)\) \(\approx\) \(0.6914426121 - 0.2627818073i\)
\(L(1)\) \(\approx\) \(0.6914426121 - 0.2627818073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.602 + 0.798i)T \)
3 \( 1 + (-0.445 - 0.895i)T \)
5 \( 1 + (-0.0922 - 0.995i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
11 \( 1 + (0.602 - 0.798i)T \)
13 \( 1 + (0.932 - 0.361i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (-0.602 - 0.798i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (0.850 - 0.526i)T \)
37 \( 1 + (-0.739 + 0.673i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (-0.739 - 0.673i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.445 + 0.895i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (-0.982 + 0.183i)T \)
67 \( 1 + (-0.932 - 0.361i)T \)
71 \( 1 + (-0.0922 + 0.995i)T \)
73 \( 1 + (-0.0922 + 0.995i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (0.932 - 0.361i)T \)
89 \( 1 + (0.273 - 0.961i)T \)
97 \( 1 + (-0.982 - 0.183i)T \)
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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

−29.86160325635515391193987122460, −28.5653941397133709597560524187, −27.870668467465771966980894982235, −26.96335853513541663513596372811, −26.267087064436922423375378769396, −25.09096449491161277465827957682, −23.16377626497332132003065177839, −22.394656501906455257121672757559, −21.42808578388034218078873982900, −20.66152252006011211731437958898, −19.437559704217151527852927155088, −17.96361839636685780702253575656, −17.723102573762924683189918347680, −16.16846304904207380705537413975, −15.0516841481330207282360591332, −13.868543881116241313282236729882, −11.85426757317824707404411375921, −11.39868460653440031300709685371, −10.32432565384692463510368718888, −9.32473300815379059326727543468, −8.01362740767093178698029352110, −6.4140749357463790794358274055, −4.56423436438192528555352392102, −3.478895469589654691061138730929, −1.81586762720425291916963846693, 0.5727025622591585135618999589, 1.55787672608300950727164539752, 4.58363321576605528872293087767, 5.728068859054925264651091244467, 6.88860087804651370353302507950, 8.2684917752328518243132939936, 8.734885473355742780726340112907, 10.72908305581497007538195467424, 11.68850345962604885454894294279, 13.31508245092313383349733646617, 14.01300612416296380438493621377, 15.65053384617832972096806459745, 16.721472511324920832110147515528, 17.49704017942053934506660519610, 18.32650952179080870886844228405, 19.597551591736363406583412439757, 20.44889006810652443772699324255, 22.24257206471719205332648065439, 23.55192377423800961407563058574, 24.29396876256180048277738683822, 24.653160716157739259645886488672, 25.997943342240161914647241371695, 27.32645333542030610670390822863, 28.09004242108721431554849535904, 28.906517716370285139504294193741

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