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Properties

Label 1-3751-3751.224-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.932 + 0.361i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.998 − 0.0570i)2-s + (−0.978 − 0.207i)3-s + (0.993 + 0.113i)4-s + (−0.0665 − 0.997i)5-s + (0.964 + 0.263i)6-s + (0.797 − 0.603i)7-s + (−0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.00951 + 0.999i)10-s + (−0.948 − 0.318i)12-s + (−0.217 + 0.976i)13-s + (−0.830 + 0.556i)14-s + (−0.142 + 0.989i)15-s + (0.974 + 0.226i)16-s + (0.723 − 0.690i)17-s + (−0.888 − 0.458i)18-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (−0.978 − 0.207i)3-s + (0.993 + 0.113i)4-s + (−0.0665 − 0.997i)5-s + (0.964 + 0.263i)6-s + (0.797 − 0.603i)7-s + (−0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.00951 + 0.999i)10-s + (−0.948 − 0.318i)12-s + (−0.217 + 0.976i)13-s + (−0.830 + 0.556i)14-s + (−0.142 + 0.989i)15-s + (0.974 + 0.226i)16-s + (0.723 − 0.690i)17-s + (−0.888 − 0.458i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.932 + 0.361i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.932 + 0.361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06054863120 - 0.3236161916i\)
\(L(\frac12)\) \(\approx\) \(-0.06054863120 - 0.3236161916i\)
\(L(1)\) \(\approx\) \(0.4761896059 - 0.2032161891i\)
\(L(1)\) \(\approx\) \(0.4761896059 - 0.2032161891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.998 - 0.0570i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.0665 - 0.997i)T \)
7 \( 1 + (0.797 - 0.603i)T \)
13 \( 1 + (-0.217 + 0.976i)T \)
17 \( 1 + (0.723 - 0.690i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (0.516 - 0.856i)T \)
29 \( 1 + (-0.0285 + 0.999i)T \)
37 \( 1 + (0.861 - 0.508i)T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.969 + 0.244i)T \)
47 \( 1 + (-0.254 + 0.967i)T \)
53 \( 1 + (-0.905 + 0.424i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.723 + 0.690i)T \)
73 \( 1 + (-0.999 - 0.0190i)T \)
79 \( 1 + (0.999 + 0.0380i)T \)
83 \( 1 + (0.123 + 0.992i)T \)
89 \( 1 + (-0.564 + 0.825i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−18.72530528948261532819858908227, −18.27905969890225871387023794068, −17.62908705852958900459199651355, −17.17984039759436475191019925334, −16.44071853070520687296478751279, −15.5697419259431379976542880932, −14.943778625296094912965467137194, −14.800434299063399246095756210750, −13.43838841561510008918160914606, −12.37650904831975977609366646374, −11.77090925566527171604326564259, −11.342680807058432255848850511216, −10.55951910547670020049513332324, −10.07149510776658367882492198806, −9.52383633241422339919861328488, −8.22289354534835218329640896647, −7.86172091667281552313423483377, −7.09129158145074752006491471155, −6.17244522200864926589401995069, −5.74898668873922869852488770676, −5.00830750019062511119852887718, −3.68648022025443140111569945166, −2.99432914247837818769007530799, −1.894253529017618290815298280572, −1.22845044698180416802727339937, 0.170235168134779342548997338805, 1.15978674027901864267033784658, 1.51845985783667068024472564623, 2.64269021072706959254656210824, 3.97513983322970335378183435511, 4.86228946965071953511603792642, 5.23957410140970572349878456007, 6.39397121329857761193954941690, 7.03824696968935580784854763128, 7.64627791184725739132301266621, 8.38126612827206243445467402657, 9.25247687217876263828445444618, 9.746847472632051737935264482088, 10.7725428529234901808264143105, 11.18122566884170942540733227071, 11.90544582005145527847359581458, 12.37298336508494974322569252948, 13.22886277195069681331924566708, 14.09766162491075765763365632035, 15.00549782011702284493383218913, 15.96399286194711239226014414263, 16.48238605269317905535881782525, 16.87329202920838050270521391572, 17.46992879229943872213244645082, 18.13128314658650568554290543582

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