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Properties

Label 1-3503-3503.30-r1-0-0
Degree $1$
Conductor $3503$
Sign $0.877 - 0.480i$
Analytic cond. $376.449$
Root an. cond. $376.449$
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  + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (0.222 − 0.974i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (0.222 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $0.877 - 0.480i$
Analytic conductor: \(376.449\)
Root analytic conductor: \(376.449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (1:\ ),\ 0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.272152038 - 0.3256289506i\)
\(L(\frac12)\) \(\approx\) \(1.272152038 - 0.3256289506i\)
\(L(1)\) \(\approx\) \(0.7749391945 + 0.3989763260i\)
\(L(1)\) \(\approx\) \(0.7749391945 + 0.3989763260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.900 + 0.433i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−18.46758367410165532574085669579, −17.81828788491878079446886782522, −17.67191858128149802809346115313, −16.82108463856223749835074761445, −16.31757775581696532224469302860, −14.890216170804574413148064977358, −14.35457499991803229590883958616, −13.46826321272869993561017279954, −12.82377739011074317689251721927, −12.23917265029896616504995483573, −11.89427757191259265912063979501, −11.16075210897838125093779103133, −10.11683976975736710841211989514, −9.73575840701671783925000824871, −8.72857311977986231615701301966, −8.16033508889362042605873216875, −7.54348386945909668141962055050, −6.27644706567788908837700800047, −5.54698168570578757524921490145, −5.05214363367572557812063774908, −4.22888724491800652437986606600, −3.05665393093055949380732494820, −2.083848550560623699447886158532, −1.48740806452266545843993913187, −1.05943560151455409170922856397, 0.281831574679902384057156353576, 1.009200293453759052483156857313, 2.28320243133826661616440807819, 3.5759839956088976880894771146, 4.223411708812182681927334609592, 4.9783883313315925395602665534, 5.60136117179497302945660941682, 6.6269819843053696894874609421, 6.80765906188946418756948238510, 7.70127198189264837817867313915, 8.77652861516596279580760389894, 9.495103396069222727691611258247, 9.937718530526644434739975551501, 10.66357293158250098276821450358, 11.45435190771926241716427244705, 11.98281927329844692776835516926, 13.46853344581227377049449545968, 13.913148355725435981884404744534, 14.55734674193090158501571796584, 14.93069071772522859890639019222, 15.98851877047432640043912315527, 16.49085160114251638056602858060, 17.18271925342339329670482330516, 17.54579560496736402757672561688, 18.24075661950843594643767061494

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