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Properties

Label 1-43e2-1849.732-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.622 + 0.782i$
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  + (−0.181 − 0.983i)2-s + (−0.322 − 0.946i)3-s + (−0.934 + 0.357i)4-s + (−0.581 − 0.813i)5-s + (−0.872 + 0.489i)6-s + (0.905 − 0.424i)7-s + (0.520 + 0.853i)8-s + (−0.791 + 0.611i)9-s + (−0.694 + 0.719i)10-s + (0.252 + 0.967i)11-s + (0.639 + 0.768i)12-s + (−0.872 − 0.489i)13-s + (−0.581 − 0.813i)14-s + (−0.581 + 0.813i)15-s + (0.744 − 0.667i)16-s + (0.957 + 0.288i)17-s + ⋯
L(s)  = 1  + (−0.181 − 0.983i)2-s + (−0.322 − 0.946i)3-s + (−0.934 + 0.357i)4-s + (−0.581 − 0.813i)5-s + (−0.872 + 0.489i)6-s + (0.905 − 0.424i)7-s + (0.520 + 0.853i)8-s + (−0.791 + 0.611i)9-s + (−0.694 + 0.719i)10-s + (0.252 + 0.967i)11-s + (0.639 + 0.768i)12-s + (−0.872 − 0.489i)13-s + (−0.581 − 0.813i)14-s + (−0.581 + 0.813i)15-s + (0.744 − 0.667i)16-s + (0.957 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (732, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3385433506 - 0.7019026389i\)
\(L(\frac12)\) \(\approx\) \(-0.3385433506 - 0.7019026389i\)
\(L(1)\) \(\approx\) \(0.4190233193 - 0.6130142142i\)
\(L(1)\) \(\approx\) \(0.4190233193 - 0.6130142142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.181 - 0.983i)T \)
3 \( 1 + (-0.322 - 0.946i)T \)
5 \( 1 + (-0.581 - 0.813i)T \)
7 \( 1 + (0.905 - 0.424i)T \)
11 \( 1 + (0.252 + 0.967i)T \)
13 \( 1 + (-0.872 - 0.489i)T \)
17 \( 1 + (0.957 + 0.288i)T \)
19 \( 1 + T \)
23 \( 1 + (0.639 - 0.768i)T \)
29 \( 1 + (-0.791 - 0.611i)T \)
31 \( 1 + (-0.872 + 0.489i)T \)
37 \( 1 + (0.989 - 0.145i)T \)
41 \( 1 + (-0.791 + 0.611i)T \)
47 \( 1 + (-0.791 - 0.611i)T \)
53 \( 1 + (0.520 - 0.853i)T \)
59 \( 1 + (-0.457 - 0.889i)T \)
61 \( 1 + (0.905 - 0.424i)T \)
67 \( 1 + (-0.181 - 0.983i)T \)
71 \( 1 + (-0.791 + 0.611i)T \)
73 \( 1 + (-0.934 - 0.357i)T \)
79 \( 1 + (-0.0365 - 0.999i)T \)
83 \( 1 + (0.905 - 0.424i)T \)
89 \( 1 + (-0.976 + 0.217i)T \)
97 \( 1 + (-0.872 + 0.489i)T \)
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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

−20.73614391505010944760873311259, −19.70516000461408620297131975971, −18.89178986843935821666311603707, −18.31465270619605127558425239152, −17.566634256806678343259685809359, −16.657011815244870311046709335516, −16.3258584833145670466657114608, −15.37243040702134164155357287855, −14.783760010361044760503451599502, −14.41799945769100044498485515706, −13.65591712514576720572216619662, −12.18190794251873302045652641307, −11.473043862651713588191799237716, −10.94653830604901357340760842569, −9.91822596353918913270765460639, −9.28857959387266551003242142447, −8.49967326003785611144465605679, −7.59261810336087737982931949547, −7.06253750021050308496208303150, −5.775811493724790947628375378653, −5.4541365431236517051448225103, −4.51352328295658062390213658333, −3.67596024198566408691494835034, −2.88299023720690189072746132030, −1.15837187093184357214188759841, 0.36793600552186067042284619945, 1.31961897414193142338119961117, 1.88497052865021363672002690265, 3.04188804392898741019523840267, 4.093059551510199146253821856090, 5.0507831360063379212553210430, 5.31626033060218787037764618103, 7.00898212245433082657999632499, 7.80678692748825838479664279884, 8.04073868654323031926047081999, 9.1427894503876500495742488074, 9.96936816653703077570478646972, 10.879325649565555471284803130031, 11.760018119222131029788449301557, 12.01577482841449018049949019428, 12.85842016728706069233044686376, 13.28836659292127010475869626800, 14.5046221972554313298517476611, 14.81053089929796785232927764789, 16.46765995893561234444474038990, 16.96152540422243045297374112120, 17.60440249188899082454446446108, 18.2254169842418524189458787105, 19.00494102446204180275239504793, 19.812210991533529508814824841905

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