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Properties

Label 1-4235-4235.1172-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.991 - 0.131i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.572 − 0.820i)2-s + (0.743 + 0.669i)3-s + (−0.345 − 0.938i)4-s + (0.974 − 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.104 + 0.994i)9-s + (0.371 − 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (0.875 + 0.483i)18-s + (0.217 + 0.976i)19-s + (−0.971 − 0.235i)23-s + (−0.548 − 0.836i)24-s + (−0.161 − 0.986i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.572 − 0.820i)2-s + (0.743 + 0.669i)3-s + (−0.345 − 0.938i)4-s + (0.974 − 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.104 + 0.994i)9-s + (0.371 − 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (0.875 + 0.483i)18-s + (0.217 + 0.976i)19-s + (−0.971 − 0.235i)23-s + (−0.548 − 0.836i)24-s + (−0.161 − 0.986i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.991 - 0.131i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.991 - 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08430774223 - 1.279151388i\)
\(L(\frac12)\) \(\approx\) \(0.08430774223 - 1.279151388i\)
\(L(1)\) \(\approx\) \(1.429256600 - 0.4327128769i\)
\(L(1)\) \(\approx\) \(1.429256600 - 0.4327128769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.572 - 0.820i)T \)
3 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (0.717 - 0.696i)T \)
17 \( 1 + (-0.508 + 0.861i)T \)
19 \( 1 + (0.217 + 0.976i)T \)
23 \( 1 + (-0.971 - 0.235i)T \)
29 \( 1 + (0.993 - 0.113i)T \)
31 \( 1 + (0.532 - 0.846i)T \)
37 \( 1 + (-0.556 - 0.830i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.875 - 0.483i)T \)
53 \( 1 + (-0.647 + 0.761i)T \)
59 \( 1 + (0.797 + 0.603i)T \)
61 \( 1 + (-0.905 - 0.424i)T \)
67 \( 1 + (-0.189 + 0.981i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (-0.475 - 0.879i)T \)
79 \( 1 + (-0.625 - 0.780i)T \)
83 \( 1 + (0.791 - 0.610i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (0.0570 + 0.998i)T \)
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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

−18.41775383471427025058155890887, −17.797747370810900555441143950527, −17.293064584639423564715254464240, −16.23210386476586022249903705436, −15.694875234844565631664166222939, −15.23145193240457472585156202351, −14.17468599085908968845221552516, −13.84441034911276258087786724546, −13.44541480754213698273498589361, −12.57818194235992940606178365511, −11.866544653482291996038703087, −11.39756223362992507002986742736, −10.06957394686833312821860347151, −9.22443079365763322117779714696, −8.61062589581764202464190573317, −8.16060623730257233559630686117, −7.14856935438907073893037667083, −6.76871347592725378088632917274, −6.17208543938122403235053327865, −5.10008355179374020149089738959, −4.42485043196600769339265912143, −3.54489232976991219098793867794, −2.90416792295468499584630738470, −2.07960641361929239463491168446, −1.00183899490985804254126395351, 0.129600017618177878446136245, 1.36637073942243049426699245308, 2.0802246678988954885550064854, 2.898952179472217656616680609627, 3.66107370447614965489907018015, 4.12145446570185209510924113338, 4.91277426033569967122735042073, 5.81419369066650279143914777637, 6.32690551511044923286049374552, 7.68085191629434883469463296366, 8.38544756584116221676363856306, 8.94428777802976658142469845460, 9.86826357029148982831576896330, 10.40752526208264572210704427675, 10.78197542018228291395794898497, 11.80472259908008392520015471895, 12.41384330639740569671153988377, 13.3286090638994021089827790811, 13.67582456931226519888281617805, 14.4525521374692401210913929814, 15.042232845494383263138301772552, 15.66384719586535478363851822680, 16.23090624615368320828926177306, 17.28107298770669006417609755801, 18.09862155253451702999000314366

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