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Properties

Label 1-21e2-441.407-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.996 - 0.0818i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.955 − 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.0747 − 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (−0.623 + 0.781i)17-s + 19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.955 − 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.0747 − 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (−0.623 + 0.781i)17-s + 19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.996 - 0.0818i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.996 - 0.0818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04351565344 - 1.061752738i\)
\(L(\frac12)\) \(\approx\) \(0.04351565344 - 1.061752738i\)
\(L(1)\) \(\approx\) \(0.6371428269 - 0.5047443137i\)
\(L(1)\) \(\approx\) \(0.6371428269 - 0.5047443137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−24.18016202221863268446308485608, −23.43591518932233965901850128866, −22.79751073420372610204322412985, −22.12415634845997290566701262160, −20.75496999058893738287650806200, −19.87207885082980275522724288152, −18.79666245270719466019300484750, −18.24385497946056625204502644416, −17.29197683364650613747832662417, −16.19167858225811446788763077439, −15.64560806747942251219878360130, −14.89227717472662060309391711944, −13.947191431018066111624967498147, −13.04032432669636322438858659625, −11.9261365362328461868028024790, −11.00300034899945062119602522850, −9.7211416175046216323830030622, −8.908185025829633185798103630287, −7.8148858775352575783270912418, −7.145665979165243859171793774869, −6.30929022587873367947158685671, −4.896580628723616862942169194007, −4.23544750703217047627562443255, −3.00238659343615493519880512165, −1.07160273334393732845546475754, 0.38408496904683329420989607489, 1.32329394277792100540273209510, 3.01332555097252680675225946524, 3.64028323779297293403295142945, 4.74236165781485115000409699498, 5.80062491661372349831937098652, 7.36425849349069067937848174481, 8.5519707884849627884251416695, 8.81302581979072367835202277999, 10.37458815382716815729470844693, 11.04946261889452908580241816114, 11.78843204749912840411453770551, 12.78630175927528238879672674684, 13.46204571952985239390817852651, 14.54475104142993600655476199364, 15.68072329383281409553716153882, 16.46961210033993033108282291466, 17.61492454984897083910337780590, 18.47710589054057152624298964653, 19.30584112473655079454863339576, 19.94021667857761444799673456820, 20.75281054346434981287106338883, 21.60737482527801852146972404896, 22.49927348196562877467132137120, 23.33990782532882564676375060807

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