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Properties

Label 1-99-99.70-r0-0-0
Degree $1$
Conductor $99$
Sign $0.634 - 0.773i$
Analytic cond. $0.459754$
Root an. cond. $0.459754$
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  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + 10-s + (−0.978 − 0.207i)13-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + 10-s + (−0.978 − 0.207i)13-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.634 - 0.773i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.634 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377528746 - 0.6517116807i\)
\(L(\frac12)\) \(\approx\) \(1.377528746 - 0.6517116807i\)
\(L(1)\) \(\approx\) \(1.416841049 - 0.5041090523i\)
\(L(1)\) \(\approx\) \(1.416841049 - 0.5041090523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−30.242744148037212099698409801672, −29.38569286527679216136248829552, −27.96635028694084940149288347550, −26.8582941183537119890644797689, −25.78817455033581080569448203526, −24.66627572106510030074288339842, −24.123643974556375341141115656826, −23.05080806592060919423131523345, −21.5910344476207087276997831922, −21.1363444522199497984308762877, −19.8784617500075265413951784088, −18.00609459195440345435480294884, −17.11452765210326349114967317587, −16.43620860842230782884591616418, −14.816289590516473801090034967892, −14.1644571027919096332053732923, −12.90717765104394622604932307680, −12.03698114787072635326227420597, −10.33889721785582389519340465762, −8.74139262152662269961341472208, −7.79447814249199164151403785918, −6.308574161327316487623545066214, −5.09835613298338854167833617697, −4.14803345859156898341664777388, −2.051154209272290996292707979316, 1.88929367714232888429154589465, 2.95197940313460662830944745466, 4.739662702424600013485232081806, 5.7576726298317267961441478195, 7.23629584318067271080398892429, 9.15101067051828633208494172939, 10.290847296602501101402086834792, 11.303887015975303779792391366045, 12.35458807551302811320579564571, 13.75848736241164608285438661782, 14.51245030193072393063045525729, 15.44562542583569225061583052695, 17.438106551630402706653758033029, 18.29905551307621987642521026413, 19.37339796302017605710783931932, 20.62271668008057316527692030776, 21.626312054839060827415111490664, 22.18821339452411641251636011627, 23.45563277764715802731911929905, 24.50577881176774742078602669404, 25.53315200170225504075262364690, 27.08886526230985954941660492807, 27.838949005668043398700738162956, 29.230420313774869891613547573916, 29.7960387903890266181872670475

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