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Properties

Label 1-4033-4033.1511-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.691 + 0.722i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 + 0.939i)2-s + (0.686 + 0.727i)3-s + (−0.766 − 0.642i)4-s + (0.448 − 0.893i)5-s + (−0.918 + 0.396i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.686 + 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.0581 − 0.998i)12-s + (−0.549 − 0.835i)13-s + (−0.230 − 0.973i)14-s + (0.957 − 0.286i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (0.686 + 0.727i)3-s + (−0.766 − 0.642i)4-s + (0.448 − 0.893i)5-s + (−0.918 + 0.396i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.686 + 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.0581 − 0.998i)12-s + (−0.549 − 0.835i)13-s + (−0.230 − 0.973i)14-s + (0.957 − 0.286i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6138635093 + 0.2621367486i\)
\(L(\frac12)\) \(\approx\) \(0.6138635093 + 0.2621367486i\)
\(L(1)\) \(\approx\) \(0.6241183658 + 0.4458929268i\)
\(L(1)\) \(\approx\) \(0.6241183658 + 0.4458929268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.686 + 0.727i)T \)
13 \( 1 + (-0.549 - 0.835i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.957 - 0.286i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.597 - 0.802i)T \)
53 \( 1 + (-0.893 - 0.448i)T \)
59 \( 1 + (0.957 - 0.286i)T \)
61 \( 1 + (0.802 + 0.597i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (-0.998 - 0.0581i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (-0.918 + 0.396i)T \)
97 \( 1 + (-0.998 + 0.0581i)T \)
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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

−18.40350478142445839307737541191, −17.80090186464691157434453541370, −17.190334531280629240907215516665, −16.255943133851752317086069988169, −15.495752082359045145436790999713, −14.32990928603284953092450565674, −13.97089033128489413433571392686, −13.36471184940667875177900930464, −12.91571629938690844398334149944, −12.00429773303746559910632983150, −11.294858895291858183658305460130, −10.5847210491739653550434055112, −9.91653260141527371365994179602, −9.25649400591517855618362101381, −8.67890722190205960866480319523, −7.7363376109561569614058454813, −6.99128645863749623878375081109, −6.634234108314966231160712932878, −5.49611977622321762433111096229, −4.32999364573186523173878861799, −3.50741624017555327487684662760, −2.8555115321695687378061054196, −2.38467986816154220363835456476, −1.55929128565650572804173114512, −0.41378128572535660791815271996, 0.194230912529700291969489951377, 1.64749632152523204205860979773, 2.377517110073992012452288137092, 3.44499691600655445955412445331, 4.34019423607853305897196831032, 5.063063014169838367112696142695, 5.57573366897688586410155250432, 6.29519050308707934706536614105, 7.45269158017323340835613292364, 8.01625514863446426824621654394, 8.615949070171513541159254400090, 9.44832226309261802677810938977, 9.92605276044927541053526056580, 10.09760753165430462019561569168, 11.45198284196349387867604472001, 12.58822641872384392526978335427, 13.23831068233280042968770758774, 13.48278857127477289891690541654, 14.59883438328171827013704198620, 15.21828716657238755316801367615, 15.62089466708353187771998597700, 16.30741078983322510983862081220, 16.76550761730316931389822207799, 17.70935458049306962016290559303, 18.118851455319940614154338829894

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