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Properties

Label 1-269-269.4-r0-0-0
Degree $1$
Conductor $269$
Sign $0.550 - 0.834i$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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  + (0.998 + 0.0468i)2-s + (0.388 − 0.921i)3-s + (0.995 + 0.0936i)4-s + (−0.946 − 0.322i)5-s + (0.430 − 0.902i)6-s + (0.628 + 0.777i)7-s + (0.990 + 0.140i)8-s + (−0.698 − 0.715i)9-s + (−0.930 − 0.366i)10-s + (−0.209 − 0.977i)11-s + (0.472 − 0.881i)12-s + (0.930 + 0.366i)13-s + (0.591 + 0.806i)14-s + (−0.664 + 0.747i)15-s + (0.982 + 0.186i)16-s + (0.209 − 0.977i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0468i)2-s + (0.388 − 0.921i)3-s + (0.995 + 0.0936i)4-s + (−0.946 − 0.322i)5-s + (0.430 − 0.902i)6-s + (0.628 + 0.777i)7-s + (0.990 + 0.140i)8-s + (−0.698 − 0.715i)9-s + (−0.930 − 0.366i)10-s + (−0.209 − 0.977i)11-s + (0.472 − 0.881i)12-s + (0.930 + 0.366i)13-s + (0.591 + 0.806i)14-s + (−0.664 + 0.747i)15-s + (0.982 + 0.186i)16-s + (0.209 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.550 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.076179029 - 1.117450581i\)
\(L(\frac12)\) \(\approx\) \(2.076179029 - 1.117450581i\)
\(L(1)\) \(\approx\) \(1.840258165 - 0.5841070263i\)
\(L(1)\) \(\approx\) \(1.840258165 - 0.5841070263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.998 + 0.0468i)T \)
3 \( 1 + (0.388 - 0.921i)T \)
5 \( 1 + (-0.946 - 0.322i)T \)
7 \( 1 + (0.628 + 0.777i)T \)
11 \( 1 + (-0.209 - 0.977i)T \)
13 \( 1 + (0.930 + 0.366i)T \)
17 \( 1 + (0.209 - 0.977i)T \)
19 \( 1 + (-0.513 - 0.858i)T \)
23 \( 1 + (-0.388 + 0.921i)T \)
29 \( 1 + (-0.792 + 0.610i)T \)
31 \( 1 + (0.912 - 0.409i)T \)
37 \( 1 + (-0.869 + 0.493i)T \)
41 \( 1 + (-0.998 + 0.0468i)T \)
43 \( 1 + (0.792 - 0.610i)T \)
47 \( 1 + (0.0702 + 0.997i)T \)
53 \( 1 + (-0.972 + 0.232i)T \)
59 \( 1 + (-0.995 - 0.0936i)T \)
61 \( 1 + (0.845 + 0.533i)T \)
67 \( 1 + (0.995 - 0.0936i)T \)
71 \( 1 + (-0.591 + 0.806i)T \)
73 \( 1 + (-0.300 + 0.953i)T \)
79 \( 1 + (0.163 + 0.986i)T \)
83 \( 1 + (0.0234 - 0.999i)T \)
89 \( 1 + (0.344 + 0.938i)T \)
97 \( 1 + (0.845 - 0.533i)T \)
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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

−25.99577843301047012561067726947, −24.960368174414643090144439138233, −23.74954075755303160634010698613, −23.04388854876679469394453648480, −22.500349324440626789300116654607, −21.17185707973282832674307173999, −20.604055069017795035755565317813, −19.962076738619059106841746418076, −18.91846011930386661958962737119, −17.28177530051317734705599687694, −16.29538737264517101752719428228, −15.40521590303126708388284194088, −14.79145194174743928563880190564, −14.04632395378409744696113492657, −12.821202463742710962695017174005, −11.73582852904819455156924289632, −10.62716612992833987896751907390, −10.34345978686311443253549991746, −8.31092534054113962049971473470, −7.68175879789422856722149010818, −6.31964716235086825004658959134, −4.893045461791652230202041358064, −4.06757295799290249517492728977, −3.478891796542804760213079816144, −1.945129200592844655981097469585, 1.330975446850984347228111706325, 2.708306841392677222814112800669, 3.66579664391080428908789461485, 5.037135052175373985355358742453, 6.057039942543714273356327885731, 7.20721665514897719274689539646, 8.14635834627501838019786903216, 8.905322201704234269201565783295, 11.242809085665393093276201962340, 11.533794360967725662153020610315, 12.516121963436879085287402718554, 13.49902065184676139857969160914, 14.20398672258807065575167691003, 15.376652376370816742408949602675, 15.930235750214468394937733411, 17.24953800295583863220017932229, 18.66345920337516423385568423327, 19.16079620622259493390744049612, 20.339105664413777036133302853882, 20.95326463987160435018595895340, 22.05878465654332331396724844133, 23.20187720204804670101844021489, 23.97658916500742528370446685908, 24.30106121912864773802845596909, 25.33249035029173838903359934505

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