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Properties

Label 1-109-109.8-r1-0-0
Degree $1$
Conductor $109$
Sign $0.570 - 0.820i$
Analytic cond. $11.7136$
Root an. cond. $11.7136$
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  + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $0.570 - 0.820i$
Analytic conductor: \(11.7136\)
Root analytic conductor: \(11.7136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (1:\ ),\ 0.570 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3722715383 + 0.1945433473i\)
\(L(\frac12)\) \(\approx\) \(-0.3722715383 + 0.1945433473i\)
\(L(1)\) \(\approx\) \(0.2560689786 + 0.5446246374i\)
\(L(1)\) \(\approx\) \(0.2560689786 + 0.5446246374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + iT \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.75520995186142178347564812143, −27.68919044344299601181685981518, −26.71542859939157222398770526439, −25.26348354581890796581601471232, −23.867462658868017828557107737398, −23.26405749513519027588190369545, −22.44242630944204779203820892886, −20.72898563672029916984103206369, −20.22054252719564150540670971176, −18.99098998267981812220911227446, −18.27527305133346673490466700807, −16.952643054010569671775200191577, −16.10269174287431894757266841949, −13.89422078312000724674820715589, −13.137192147324383818754801108770, −12.360885602046920440221108929589, −11.21841516029777344790724374818, −10.24809940308823090587421850579, −8.57762443096116070566551161353, −7.65929759878632599725332026010, −5.83169888885022964738773127398, −4.53912016805479367058573928114, −3.0327007543620791534218296063, −1.14193589245448378415630686584, −0.22926299491722091320617168204, 3.18135188418597310455838168252, 4.462790691293662762165704122173, 5.81460448088497803427815167541, 6.69297681830805513491866140920, 8.22742941591730032358147990330, 9.440394062286165540172039444606, 10.539089234711637474215211741392, 11.83870811066138097539338072730, 13.28337833872030741586525440777, 14.81744023663084938693727193411, 15.52364216678287891821395540270, 16.05084826721105041467767988098, 17.53589159245920384394887783822, 18.35450813804412335000760027205, 19.477601922318546575203730571443, 21.43346701548343872647663074132, 21.98100261499039511230620938297, 23.24505118937063062826413322072, 23.52248528218587462430123185304, 25.3647068447331607010925268472, 26.148078726939694279068705618859, 26.77595259758045004074072566646, 28.13934760666482267447548157159, 28.50567063201277303005862146830, 30.48203251596384573001926505089

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