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Properties

Label 1-3751-3751.340-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.955 + 0.294i$
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  + (0.654 + 0.755i)2-s − 3-s + (−0.142 + 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + 9-s + (0.142 + 0.989i)10-s + (0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + (0.654 + 0.755i)18-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s − 3-s + (−0.142 + 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + 9-s + (0.142 + 0.989i)10-s + (0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + (0.654 + 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.955 + 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3509243346 + 2.333123791i\)
\(L(\frac12)\) \(\approx\) \(0.3509243346 + 2.333123791i\)
\(L(1)\) \(\approx\) \(1.028185392 + 0.9794861356i\)
\(L(1)\) \(\approx\) \(1.028185392 + 0.9794861356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 - T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (0.142 + 0.989i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−18.162249700595113331642796011416, −17.73038875795014424705962464765, −17.178875980539396657239219620754, −16.20332981849948206328847299772, −15.54210526878501033223851237065, −14.716589469507503628943215969605, −14.10392049099548911711039176153, −13.20189003691369567269674150186, −12.74232914055934629661739140368, −12.14977616978626053256120349317, −11.39161945285236180873558019424, −10.73983221126374430719216630154, −10.3075948220403497944545938535, −9.32320577070270535527716980919, −8.839723464526815432288177903, −7.58491807464384618749053170224, −6.71387948505357642209106840298, −5.85330380551159878165968985532, −5.15155634960993661565306704615, −5.01702648760489815261678367440, −4.15141643100781009907349423779, −2.87105305258871300496225442822, −2.19604653462341091697098354216, −1.19986361509799665915221363800, −0.67114964744847502818603301369, 1.31424161002689759961578276421, 2.00873108006990493892350179486, 3.15698196376217769401365741726, 4.297146474828213873300115450107, 4.57995159783866496996645752919, 5.65210269779146968402933492896, 5.99576556315884184764170105458, 6.74327938371392921355689107795, 7.40192207375501635795920813316, 8.13222116136123957198042053583, 9.11833292217853797871753604616, 9.99491252425939835168707774030, 10.74564951934952113556562374894, 11.3966519751850319456189101434, 12.05496602579512656834096993497, 12.792354090962628992600171817203, 13.57603077922314162820756253015, 14.11374686726083750331716609185, 14.912921975190941678396097328519, 15.24898923267159002455604377183, 16.521046368888258637325636545366, 16.82082437444432674102112344692, 17.36481751177205955277373754572, 17.97022128488202265804391441540, 18.60091865911138718004725349752

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