Properties

Label 1-11e2-121.64-r0-0-0
Degree $1$
Conductor $121$
Sign $0.998 + 0.0519i$
Analytic cond. $0.561921$
Root an. cond. $0.561921$
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  + (0.941 + 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 + 0.633i)4-s + (0.974 + 0.226i)5-s + (0.610 − 0.791i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)10-s + (0.841 − 0.540i)12-s + (−0.998 − 0.0570i)13-s + (−0.921 + 0.389i)14-s + (0.516 − 0.856i)15-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (−0.564 − 0.825i)18-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 + 0.633i)4-s + (0.974 + 0.226i)5-s + (0.610 − 0.791i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)10-s + (0.841 − 0.540i)12-s + (−0.998 − 0.0570i)13-s + (−0.921 + 0.389i)14-s + (0.516 − 0.856i)15-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (−0.564 − 0.825i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.998 + 0.0519i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.998 + 0.0519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.988198536 + 0.05163234554i\)
\(L(\frac12)\) \(\approx\) \(1.988198536 + 0.05163234554i\)
\(L(1)\) \(\approx\) \(1.860211855 + 0.04070407439i\)
\(L(1)\) \(\approx\) \(1.860211855 + 0.04070407439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.941 - 0.336i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.974 - 0.226i)T \)
7 \( 1 + (0.736 - 0.676i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (0.362 + 0.931i)T \)
37 \( 1 + (0.254 - 0.967i)T \)
41 \( 1 + (0.0285 - 0.999i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.564 - 0.825i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (0.0285 + 0.999i)T \)
61 \( 1 + (-0.941 + 0.336i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.696 - 0.717i)T \)
73 \( 1 + (-0.993 - 0.113i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (0.870 + 0.491i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.974 + 0.226i)T \)
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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

−29.15283555366935325430320755953, −28.338175633679917153269679726997, −27.01637149895028378882806501502, −25.8019844156838344420251623862, −25.14826404207585249278822685560, −23.7929316575575248629774242970, −22.68460543447166350748379952731, −21.70898628958347316297489449440, −21.24919248407906334386454310520, −19.9744697884982794216060015323, −19.43238824328126823481433810953, −17.30024969493601089075439886516, −16.47928924982655461874517102319, −15.28635879398672736934914247756, −14.29539578225809172849690735913, −13.42470527987974189866326310482, −12.40912505556148934446391606590, −10.65489462850802043727304532176, −10.15999586299839869842975772423, −8.99124949338266679976341822539, −6.93740463677438208683490533274, −5.6757439077837696786050714570, −4.575583584172212447079619551899, −3.40811845011618741621217530942, −2.08836937552795403034577199294, 2.255078667453448343912549963682, 2.87880358814718829833052357426, 4.97787038141741753559196807881, 6.304625054053458086052137837230, 6.81536636173387568950392480905, 8.3539485117198093214118722933, 9.70527608184165614420147847692, 11.46074436249742873833747428172, 12.655947887885655248911835643854, 13.20525643295244625369264518736, 14.32055912756514508644521680344, 15.13519259126596100308851246338, 16.65271324996824676521313327054, 17.63849509385868320594641713261, 18.80696977524650238807714704187, 19.92663088581853518750919941070, 21.10202055165406329092317333416, 22.17489072007558572467140588536, 22.87523225055903288907029731092, 24.2361184971512463752893485101, 24.97132528213198167596062797314, 25.563248859557674135254424932565, 26.53416630915748622716618924087, 28.66919950269966130444245508610, 29.34277015549199240480799809487

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