Properties

Label 1-1183-1183.781-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.216 + 0.976i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.692 − 0.721i)2-s + (−0.919 + 0.391i)3-s + (−0.0402 − 0.999i)4-s + (0.428 + 0.903i)5-s + (−0.354 + 0.935i)6-s + (−0.748 − 0.663i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (0.692 + 0.721i)11-s + (0.428 + 0.903i)12-s + (−0.748 − 0.663i)15-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (−0.0402 − 0.999i)18-s + (−0.5 + 0.866i)19-s + (0.885 − 0.464i)20-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (−0.919 + 0.391i)3-s + (−0.0402 − 0.999i)4-s + (0.428 + 0.903i)5-s + (−0.354 + 0.935i)6-s + (−0.748 − 0.663i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (0.692 + 0.721i)11-s + (0.428 + 0.903i)12-s + (−0.748 − 0.663i)15-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (−0.0402 − 0.999i)18-s + (−0.5 + 0.866i)19-s + (0.885 − 0.464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9133589702 + 0.7329580328i\)
\(L(\frac12)\) \(\approx\) \(0.9133589702 + 0.7329580328i\)
\(L(1)\) \(\approx\) \(1.076752469 + 0.001462760818i\)
\(L(1)\) \(\approx\) \(1.076752469 + 0.001462760818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.692 - 0.721i)T \)
3 \( 1 + (-0.919 + 0.391i)T \)
5 \( 1 + (0.428 + 0.903i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (-0.200 + 0.979i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (0.987 - 0.160i)T \)
37 \( 1 + (-0.632 - 0.774i)T \)
41 \( 1 + (0.120 - 0.992i)T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (-0.0402 + 0.999i)T \)
53 \( 1 + (-0.200 + 0.979i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (-0.200 - 0.979i)T \)
67 \( 1 + (-0.845 - 0.534i)T \)
71 \( 1 + (0.120 - 0.992i)T \)
73 \( 1 + (0.692 + 0.721i)T \)
79 \( 1 + (-0.0402 + 0.999i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.35688718099810680782769440265, −20.51205604202867782197052627850, −19.564507576371284009096057010, −18.41886953175831487038442511832, −17.729127913331413168058019601625, −16.99164494351802189125748224573, −16.47424860120013948941424773647, −15.93163368458182902868298976791, −14.86544300915292621141202474162, −13.75899747209527404494106259532, −13.375837247056538503417364157886, −12.57151617696843833425522815024, −11.79529897725921986954867393029, −11.25792627749719353833609205963, −9.94454425654823371080925975226, −8.88893620332723865020594149753, −8.23570114745073125736443471470, −7.11199002139418841690220625899, −6.38681701028222062874636133167, −5.77028922326355362542318449525, −4.80519755014660209992107806036, −4.404645209384094844127051779801, −2.97119244183597025121931991364, −1.727170952318864538857112233882, −0.40970561100261543379603790276, 1.46953163221037341062309630241, 2.165601502447432848524597050560, 3.64990706175828975671275117955, 3.998636348056008351909275038394, 5.160424516661825466611667410928, 6.05364611017490893841197588170, 6.44406230588098185125381647149, 7.50879419484181598235983998939, 9.20703128593574249815906935272, 9.84588098337350373506772428160, 10.58917101666215895749379636420, 11.06662060862171813665144830163, 12.07564356019854835568759966675, 12.49654774136279108147058915732, 13.604396889106565426824120188767, 14.3803907067448162423510630548, 15.17464318637712211032388288641, 15.61590109748814656995841091804, 17.048432657972703737937938489, 17.47874442528775127321520293271, 18.43205784622225979931314935806, 19.06734747415019441062868657245, 19.93316422249442003472552513757, 21.00930222056390549445154776877, 21.40033008115241825347206936373

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