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Properties

Label 1-43e2-1849.646-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.132 + 0.991i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
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  + (0.905 − 0.424i)2-s + (−0.457 − 0.889i)3-s + (0.639 − 0.768i)4-s + (0.520 + 0.853i)5-s + (−0.791 − 0.611i)6-s + (0.109 + 0.994i)7-s + (0.252 − 0.967i)8-s + (−0.581 + 0.813i)9-s + (0.833 + 0.551i)10-s + (−0.322 + 0.946i)11-s + (−0.976 − 0.217i)12-s + (−0.791 + 0.611i)13-s + (0.520 + 0.853i)14-s + (0.520 − 0.853i)15-s + (−0.181 − 0.983i)16-s + (−0.997 − 0.0729i)17-s + ⋯
L(s)  = 1  + (0.905 − 0.424i)2-s + (−0.457 − 0.889i)3-s + (0.639 − 0.768i)4-s + (0.520 + 0.853i)5-s + (−0.791 − 0.611i)6-s + (0.109 + 0.994i)7-s + (0.252 − 0.967i)8-s + (−0.581 + 0.813i)9-s + (0.833 + 0.551i)10-s + (−0.322 + 0.946i)11-s + (−0.976 − 0.217i)12-s + (−0.791 + 0.611i)13-s + (0.520 + 0.853i)14-s + (0.520 − 0.853i)15-s + (−0.181 − 0.983i)16-s + (−0.997 − 0.0729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (646, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9623198109 + 0.8418249567i\)
\(L(\frac12)\) \(\approx\) \(0.9623198109 + 0.8418249567i\)
\(L(1)\) \(\approx\) \(1.298871016 - 0.1585336043i\)
\(L(1)\) \(\approx\) \(1.298871016 - 0.1585336043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.905 - 0.424i)T \)
3 \( 1 + (-0.457 - 0.889i)T \)
5 \( 1 + (0.520 + 0.853i)T \)
7 \( 1 + (0.109 + 0.994i)T \)
11 \( 1 + (-0.322 + 0.946i)T \)
13 \( 1 + (-0.791 + 0.611i)T \)
17 \( 1 + (-0.997 - 0.0729i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.976 + 0.217i)T \)
29 \( 1 + (-0.581 - 0.813i)T \)
31 \( 1 + (-0.791 - 0.611i)T \)
37 \( 1 + (-0.0365 - 0.999i)T \)
41 \( 1 + (-0.581 + 0.813i)T \)
47 \( 1 + (-0.581 - 0.813i)T \)
53 \( 1 + (0.252 + 0.967i)T \)
59 \( 1 + (-0.872 + 0.489i)T \)
61 \( 1 + (0.109 + 0.994i)T \)
67 \( 1 + (0.905 - 0.424i)T \)
71 \( 1 + (-0.581 + 0.813i)T \)
73 \( 1 + (0.639 + 0.768i)T \)
79 \( 1 + (0.391 + 0.920i)T \)
83 \( 1 + (0.109 + 0.994i)T \)
89 \( 1 + (0.744 + 0.667i)T \)
97 \( 1 + (-0.791 - 0.611i)T \)
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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

−20.37238063427276327219552199204, −19.76477444092511955786264814672, −18.032626047769666678933690396264, −17.45593455196318675774275817624, −16.81736091658835167011240601655, −16.18960650859903971527545503280, −15.79913850351928732059491861989, −14.736047608139687644806605318207, −14.03393551019094434058588585993, −13.42164379794408669216251365613, −12.64980689293778694262583254092, −11.84890824871740029968764877584, −10.999465893047078393565786450310, −10.38859723837188015012348081983, −9.47964460449426773308730482498, −8.5434821286294431873091248443, −7.77725094688849580833502714325, −6.722304641080974913223003119188, −5.91401550130285885097491287278, −5.08749177898923552485011968831, −4.78736188980740192678564620433, −3.74022997705150144714176244740, −3.10974100372997450271285208092, −1.75813362283578204687669831072, −0.29677086912538855576467156840, 1.6597593089028735601040768765, 2.25941316276963859000450843504, 2.64067206904547425788328980541, 4.018056593928226742345785205878, 5.121685230924574974696873429660, 5.638170617197472533272917617408, 6.4418124817374882013716684199, 7.110177271062677677996420183929, 7.77579599654414468097185902209, 9.32286542707797223322962679424, 9.88181409034326971773825551384, 10.90786102491351297210002427020, 11.60597123629561508544291884330, 12.056517331350161152327551479217, 12.882058477385713999606766401985, 13.55323237896049489377246279920, 14.25569625012789099814400799062, 14.9745960224158527570238718566, 15.58948757932854246067723830749, 16.63461511140998488590272338723, 17.68476546468794529535421852242, 18.27227022426908187787511610180, 18.69722123831176374960020259250, 19.65836589489502573342179148281, 20.1770476817084124781693594844

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