Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (4032, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(5.681272646\)
\(L(\frac12,\chi)\)  \(\approx\)  \(5.681272646\)
\(L(\chi,1)\)  \(\approx\)  \(2.855239499\)
\(L(1,\chi)\)  \(\approx\)  \(2.855239499\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.57461442709341223998967133665, −18.12867267955007496081691410066, −16.65285567440518928114386514807, −16.25610147994830901209726597634, −15.33670521593108582187643860710, −15.12490014997311251098790858583, −14.43034712197013933097609669088, −13.724099892511429493235984951224, −13.12810690429612653785251464531, −12.46475010014542513263326283844, −11.651106191131539592653562492169, −11.035886439707883637094400585366, −10.47454557221647823609131398065, −9.40323449567117887585132291395, −8.37274791795417712584987878569, −7.88640867288464697636139522933, −7.45712642343535876437049664941, −6.66611406697288217421303426204, −5.2729618688142222906983841244, −5.08253018871779565846168282744, −4.018205732080327339231579514898, −3.40961645693790757429275014625, −2.95312144487582789266286814231, −1.80929254399465482209805615269, −1.16558507272148551350484373853, 1.16558507272148551350484373853, 1.80929254399465482209805615269, 2.95312144487582789266286814231, 3.40961645693790757429275014625, 4.018205732080327339231579514898, 5.08253018871779565846168282744, 5.2729618688142222906983841244, 6.66611406697288217421303426204, 7.45712642343535876437049664941, 7.88640867288464697636139522933, 8.37274791795417712584987878569, 9.40323449567117887585132291395, 10.47454557221647823609131398065, 11.035886439707883637094400585366, 11.651106191131539592653562492169, 12.46475010014542513263326283844, 13.12810690429612653785251464531, 13.724099892511429493235984951224, 14.43034712197013933097609669088, 15.12490014997311251098790858583, 15.33670521593108582187643860710, 16.25610147994830901209726597634, 16.65285567440518928114386514807, 18.12867267955007496081691410066, 18.57461442709341223998967133665

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