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Properties

Label 1-4763-4763.1091-r1-0-0
Degree $1$
Conductor $4763$
Sign $0.266 - 0.963i$
Analytic cond. $511.855$
Root an. cond. $511.855$
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  + (0.829 − 0.559i)2-s + (−0.780 + 0.624i)3-s + (0.374 − 0.927i)4-s + (−0.0407 + 0.999i)5-s + (−0.297 + 0.954i)6-s + (−0.401 + 0.915i)7-s + (−0.207 − 0.978i)8-s + (0.219 − 0.975i)9-s + (0.524 + 0.851i)10-s + (0.286 + 0.957i)12-s + (0.162 − 0.986i)13-s + (0.179 + 0.983i)14-s + (−0.592 − 0.805i)15-s + (−0.719 − 0.694i)16-s + (−0.795 − 0.606i)17-s + (−0.363 − 0.931i)18-s + ⋯
L(s)  = 1  + (0.829 − 0.559i)2-s + (−0.780 + 0.624i)3-s + (0.374 − 0.927i)4-s + (−0.0407 + 0.999i)5-s + (−0.297 + 0.954i)6-s + (−0.401 + 0.915i)7-s + (−0.207 − 0.978i)8-s + (0.219 − 0.975i)9-s + (0.524 + 0.851i)10-s + (0.286 + 0.957i)12-s + (0.162 − 0.986i)13-s + (0.179 + 0.983i)14-s + (−0.592 − 0.805i)15-s + (−0.719 − 0.694i)16-s + (−0.795 − 0.606i)17-s + (−0.363 − 0.931i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4763\)    =    \(11 \cdot 433\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(511.855\)
Root analytic conductor: \(511.855\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4763} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4763,\ (1:\ ),\ 0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7735912789 - 0.5885549318i\)
\(L(\frac12)\) \(\approx\) \(0.7735912789 - 0.5885549318i\)
\(L(1)\) \(\approx\) \(1.031068148 + 0.006680349724i\)
\(L(1)\) \(\approx\) \(1.031068148 + 0.006680349724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
433 \( 1 \)
good2 \( 1 + (0.829 - 0.559i)T \)
3 \( 1 + (-0.780 + 0.624i)T \)
5 \( 1 + (-0.0407 + 0.999i)T \)
7 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (0.162 - 0.986i)T \)
17 \( 1 + (-0.795 - 0.606i)T \)
19 \( 1 + (-0.805 - 0.592i)T \)
23 \( 1 + (-0.369 + 0.929i)T \)
29 \( 1 + (0.937 + 0.347i)T \)
31 \( 1 + (0.433 + 0.901i)T \)
37 \( 1 + (-0.529 + 0.848i)T \)
41 \( 1 + (0.947 - 0.320i)T \)
43 \( 1 + (-0.906 + 0.422i)T \)
47 \( 1 + (-0.167 + 0.985i)T \)
53 \( 1 + (-0.677 + 0.735i)T \)
59 \( 1 + (0.947 + 0.320i)T \)
61 \( 1 + (-0.412 - 0.911i)T \)
67 \( 1 + (0.422 + 0.906i)T \)
71 \( 1 + (-0.739 - 0.673i)T \)
73 \( 1 + (0.544 + 0.838i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (0.976 + 0.213i)T \)
89 \( 1 + (-0.573 + 0.819i)T \)
97 \( 1 + (0.981 + 0.190i)T \)
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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

−17.70916541022951537930519737384, −17.25039527417330119267438676883, −16.68448124960646935378801758705, −16.26080834837973586815162999758, −15.71118267165622354283804538769, −14.60941498584470586266675814174, −13.93847754584226743216477249021, −13.12723534670155397327815483365, −13.06270902951907046680082849269, −12.12869211768749431118827787085, −11.72741000205586378474130397458, −10.83157148000240510626226320836, −10.17142135285102758361837857986, −9.00158359956355809690323212767, −8.26028162866710898087073376315, −7.73739309774904273125141335861, −6.700443005072226682298239837826, −6.46222009284615092922609285537, −5.74118767583389386745911356193, −4.73912017986526010503989475269, −4.32341043974578208477834696791, −3.75893050424422121932818853989, −2.28687312733427400536688458981, −1.72087439576446275626403155053, −0.55226359344255807265634335302, 0.1733725282181509904954663219, 1.333972922772825605183077326008, 2.54149015293064257970924429214, 2.975254764317126176302531445961, 3.678113055208614223374531078324, 4.61731721948927314319709916810, 5.22636143657015927485130916928, 5.98156060573009195178957352105, 6.496754165041559011337260347446, 7.05832841260744441307412543154, 8.37209673318239105423471940884, 9.34666287184291363963014982434, 9.9114443922010852242808944793, 10.60944382575572623627259998572, 11.09110210032985644772744765236, 11.717730294051246297947840110394, 12.33423980902123327731910290671, 13.007599627063812704346887810040, 13.79365742784124716183231352281, 14.60671171285456505160433175958, 15.258459047364269904907266319936, 15.72965864462947786464167356721, 15.98039927501568321851926496139, 17.405713277978272255861909042319, 17.89337168537662071818492482953

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