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Properties

Label 1-4033-4033.1002-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.692 - 0.721i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.173 + 0.984i)2-s + (0.893 + 0.448i)3-s + (−0.939 − 0.342i)4-s + (−0.597 + 0.802i)5-s + (−0.597 + 0.802i)6-s + (0.396 + 0.918i)7-s + (0.5 − 0.866i)8-s + (0.597 + 0.802i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.686 − 0.727i)12-s + (0.993 + 0.116i)13-s + (−0.973 + 0.230i)14-s + (−0.893 + 0.448i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.893 + 0.448i)3-s + (−0.939 − 0.342i)4-s + (−0.597 + 0.802i)5-s + (−0.597 + 0.802i)6-s + (0.396 + 0.918i)7-s + (0.5 − 0.866i)8-s + (0.597 + 0.802i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.686 − 0.727i)12-s + (0.993 + 0.116i)13-s + (−0.973 + 0.230i)14-s + (−0.893 + 0.448i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7688579896 + 1.804164832i\)
\(L(\frac12)\) \(\approx\) \(-0.7688579896 + 1.804164832i\)
\(L(1)\) \(\approx\) \(0.5803551863 + 1.061040636i\)
\(L(1)\) \(\approx\) \(0.5803551863 + 1.061040636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (-0.597 + 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.686 + 0.727i)T \)
13 \( 1 + (0.993 + 0.116i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (-0.893 + 0.448i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (-0.686 - 0.727i)T \)
59 \( 1 + (-0.893 + 0.448i)T \)
61 \( 1 + (-0.973 + 0.230i)T \)
67 \( 1 + (0.973 + 0.230i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (0.0581 + 0.998i)T \)
97 \( 1 + (0.0581 + 0.998i)T \)
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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

−18.30737565673670550712697018464, −17.56015360282704769230433948272, −16.83681180550267545040012684832, −15.90934508776450085078642091841, −15.4433364022017868828258132888, −14.17674866840195606590055680668, −13.795395313899394019547009275, −13.1283018000384586552789094656, −12.8361420926980007337860386015, −11.72937850270867177842365470248, −11.21455670099321542519938547866, −10.637191058980895140068614557245, −9.47852495908026825372554618331, −9.08732058114474817511643258984, −8.36424078457376298833245404707, −7.6979578495168347125891530687, −7.33449968669822808331991192053, −5.9566347563858989279619618620, −4.78337290860436735303832569932, −4.43114076175869012985271744866, −3.3423616005914521912244275603, −3.09513633201954688215001555387, −1.94226365718652413473194008857, −0.944412003063396412196335122262, −0.66256353727911702967206695641, 1.41569950019768270338809273351, 2.31523929007837381594514921289, 3.31632802916057765000569800659, 3.8992142763323072496523076309, 4.76121213950138142873352676712, 5.4744786696516462838287022250, 6.34449651827017712632965113057, 7.19742112017636328461917429224, 7.78429833092643128150124089705, 8.4601481740954836569579937940, 8.87149855690468882124702130870, 9.75223336094274623043009428215, 10.613542638780569987007677592, 10.92307890383961304343438914046, 12.332046460442677860737903118932, 12.81126893117872618611836222405, 13.82439122754854559149782807832, 14.40942844258543444488424773275, 14.941128370693975210884312226501, 15.45877420186183020604954578998, 15.88022580692169212838464328967, 16.576384834856095334921325763433, 17.71605254621474019293892973713, 18.29886912215070233739636077426, 18.73119312215813334964766202594

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