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Properties

Label 1-3332-3332.2923-r1-0-0
Degree $1$
Conductor $3332$
Sign $-0.672 + 0.740i$
Analytic cond. $358.073$
Root an. cond. $358.073$
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  + (0.826 + 0.563i)3-s + (−0.0747 + 0.997i)5-s + (0.365 + 0.930i)9-s + (0.365 − 0.930i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.5 − 0.866i)19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (0.222 + 0.974i)29-s + (−0.5 − 0.866i)31-s + (0.826 − 0.563i)33-s + (−0.955 − 0.294i)37-s + (0.0747 + 0.997i)39-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)3-s + (−0.0747 + 0.997i)5-s + (0.365 + 0.930i)9-s + (0.365 − 0.930i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.5 − 0.866i)19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (0.222 + 0.974i)29-s + (−0.5 − 0.866i)31-s + (0.826 − 0.563i)33-s + (−0.955 − 0.294i)37-s + (0.0747 + 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(358.073\)
Root analytic conductor: \(358.073\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3332,\ (1:\ ),\ -0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218106272 + 2.751723287i\)
\(L(\frac12)\) \(\approx\) \(1.218106272 + 2.751723287i\)
\(L(1)\) \(\approx\) \(1.318822947 + 0.6073960206i\)
\(L(1)\) \(\approx\) \(1.318822947 + 0.6073960206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.826 + 0.563i)T \)
5 \( 1 + (-0.0747 + 0.997i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.955 - 0.294i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 - T \)
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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

−18.34940304484581828034330966495, −17.79564483398594196729266581072, −17.210873511204681893359409591670, −16.23831390397263232135906712345, −15.555619091130322421016882183594, −15.060770096485996856193687535, −13.98913827497130164410647453056, −13.68096823766124792839477310523, −12.755453612582986353587944686842, −12.28384473705104713555394505624, −11.796491637343607512280916477810, −10.510435393682667246274415904817, −9.760831644070282206044645753974, −9.12030694479750979326106167644, −8.4696847678219152693647843563, −7.72042294931954208042378664089, −7.29552678249621660356086836418, −6.105213408080666984657938465937, −5.55423032044772550289089280661, −4.39804607205622889291815178531, −3.84226406774763506216563924197, −2.96719020636128739198712022682, −1.84363170970238104837173773031, −1.37394835078529829948422722445, −0.42426720361439241899988354514, 0.96949345499001001935529738991, 2.1933354803057867712821215817, 2.72440634163228187679433115537, 3.78492304750925274852847930168, 3.92646878792673910338905378216, 5.16904115875109012673742029546, 6.06159907839740733399584198462, 6.85452543699393621713505382457, 7.52703740071133868269814388053, 8.42352841138611773702846848826, 9.02764043216605833918463962819, 9.66055973564143971486681536280, 10.642698679368227298002343648947, 11.001888187029703302129837010175, 11.717783536338453807171032113129, 12.80253267805897245773539011709, 13.78174249636805193341613997164, 14.0838801421800144260570980788, 14.59690337643321909488299292781, 15.568055102528178393798055816866, 15.968701157351161569337229812308, 16.659473222067454536149932304006, 17.62550377135729206881973800573, 18.57375033179188410978466490559, 18.82796590986870160416931582444

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