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Properties

Label 1-3751-3751.1345-r1-0-0
Degree $1$
Conductor $3751$
Sign $-0.969 - 0.243i$
Analytic cond. $403.100$
Root an. cond. $403.100$
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  + (−0.142 + 0.989i)2-s + (0.978 + 0.207i)3-s + (−0.959 − 0.281i)4-s + (0.953 + 0.299i)5-s + (−0.345 + 0.938i)6-s + (0.0475 + 0.998i)7-s + (0.415 − 0.909i)8-s + (0.913 + 0.406i)9-s + (−0.432 + 0.901i)10-s + (−0.879 − 0.475i)12-s + (−0.380 + 0.924i)13-s + (−0.995 − 0.0950i)14-s + (0.870 + 0.491i)15-s + (0.841 + 0.540i)16-s + (0.999 − 0.0190i)17-s + (−0.532 + 0.846i)18-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (0.978 + 0.207i)3-s + (−0.959 − 0.281i)4-s + (0.953 + 0.299i)5-s + (−0.345 + 0.938i)6-s + (0.0475 + 0.998i)7-s + (0.415 − 0.909i)8-s + (0.913 + 0.406i)9-s + (−0.432 + 0.901i)10-s + (−0.879 − 0.475i)12-s + (−0.380 + 0.924i)13-s + (−0.995 − 0.0950i)14-s + (0.870 + 0.491i)15-s + (0.841 + 0.540i)16-s + (0.999 − 0.0190i)17-s + (−0.532 + 0.846i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.969 - 0.243i$
Analytic conductor: \(403.100\)
Root analytic conductor: \(403.100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (1:\ ),\ -0.969 - 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4509708747 + 3.646384955i\)
\(L(\frac12)\) \(\approx\) \(-0.4509708747 + 3.646384955i\)
\(L(1)\) \(\approx\) \(1.056020399 + 1.179907716i\)
\(L(1)\) \(\approx\) \(1.056020399 + 1.179907716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.953 + 0.299i)T \)
7 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (-0.380 + 0.924i)T \)
17 \( 1 + (0.999 - 0.0190i)T \)
19 \( 1 + (-0.905 - 0.424i)T \)
23 \( 1 + (0.362 + 0.931i)T \)
29 \( 1 + (0.921 - 0.389i)T \)
37 \( 1 + (0.761 - 0.647i)T \)
41 \( 1 + (-0.969 - 0.244i)T \)
43 \( 1 + (0.595 - 0.803i)T \)
47 \( 1 + (0.897 - 0.441i)T \)
53 \( 1 + (-0.988 + 0.151i)T \)
59 \( 1 + (-0.969 + 0.244i)T \)
61 \( 1 + (-0.897 + 0.441i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.797 + 0.603i)T \)
73 \( 1 + (-0.0475 + 0.998i)T \)
79 \( 1 + (-0.861 + 0.508i)T \)
83 \( 1 + (-0.988 + 0.151i)T \)
89 \( 1 + (-0.0855 + 0.996i)T \)
97 \( 1 + (0.993 + 0.113i)T \)
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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

−18.37681675109128820798155730815, −17.34168019784710195374390494953, −17.09283120094510113693948041643, −16.20053654180626048342203724454, −14.94706016953105265754885813314, −14.30551430792435186722499036622, −13.96122797232452031787375548712, −13.05180811503706265352118246175, −12.770800996518741964949163553381, −12.11769379702662695391958422633, −10.74235520753007141583511294186, −10.37761196327653639395620303349, −9.79044304120934672753933986714, −9.16583334465286384580743082484, −8.18076073960660429544643089675, −7.95513505202195173323996992942, −6.85215855720103482775485545500, −5.94201034717158035300654180580, −4.77431538352508999150533740374, −4.3819747057834208675576404464, −3.195716454411555314792237896773, −2.881092976755552757283677622000, −1.837816351328572102529890185779, −1.20720398967648292370494581955, −0.48538331585458412298977848326, 1.17555638107021177720685733557, 2.06158881915145719412654511295, 2.75188125094707613860014008363, 3.73858611331000695569029701926, 4.66357501511176106338449376786, 5.35419380560136900707004163143, 6.09415375670327310404063794953, 6.85479728989221341916622879563, 7.52258940106439512942048778192, 8.42210462752755914954758504613, 8.98857519771129436981844160333, 9.52806204339532155963639153265, 10.030563787292843174021092931446, 10.944704609232855785634955256140, 12.19852746594719449396407998253, 12.85023478773103249566571641663, 13.67625928895038474314838327087, 14.15657510586159820182582909036, 14.680650215038733951214880650632, 15.37154144606647188875178697537, 15.82959361990236297119577044603, 16.875852074610069369031030632815, 17.24898049291422843819633681553, 18.22221156626150636618133520560, 18.810453707225310210264447384283

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