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Properties

Label 1-113-113.58-r1-0-0
Degree $1$
Conductor $113$
Sign $-0.900 - 0.434i$
Analytic cond. $12.1435$
Root an. cond. $12.1435$
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  + (0.433 − 0.900i)2-s + (0.815 − 0.578i)3-s + (−0.623 − 0.781i)4-s + (0.578 − 0.815i)5-s + (−0.167 − 0.985i)6-s + (0.222 + 0.974i)7-s + (−0.974 + 0.222i)8-s + (0.330 − 0.943i)9-s + (−0.483 − 0.875i)10-s + (−0.111 − 0.993i)11-s + (−0.960 − 0.276i)12-s + (0.532 − 0.846i)13-s + (0.974 + 0.222i)14-s − i·15-s + (−0.222 + 0.974i)16-s + (−0.998 − 0.0560i)17-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (0.815 − 0.578i)3-s + (−0.623 − 0.781i)4-s + (0.578 − 0.815i)5-s + (−0.167 − 0.985i)6-s + (0.222 + 0.974i)7-s + (−0.974 + 0.222i)8-s + (0.330 − 0.943i)9-s + (−0.483 − 0.875i)10-s + (−0.111 − 0.993i)11-s + (−0.960 − 0.276i)12-s + (0.532 − 0.846i)13-s + (0.974 + 0.222i)14-s − i·15-s + (−0.222 + 0.974i)16-s + (−0.998 − 0.0560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(12.1435\)
Root analytic conductor: \(12.1435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (1:\ ),\ -0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6292751168 - 2.750816027i\)
\(L(\frac12)\) \(\approx\) \(0.6292751168 - 2.750816027i\)
\(L(1)\) \(\approx\) \(1.139276203 - 1.351249516i\)
\(L(1)\) \(\approx\) \(1.139276203 - 1.351249516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (0.433 - 0.900i)T \)
3 \( 1 + (0.815 - 0.578i)T \)
5 \( 1 + (0.578 - 0.815i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (-0.111 - 0.993i)T \)
13 \( 1 + (0.532 - 0.846i)T \)
17 \( 1 + (-0.998 - 0.0560i)T \)
19 \( 1 + (-0.167 + 0.985i)T \)
23 \( 1 + (0.985 - 0.167i)T \)
29 \( 1 + (-0.0560 + 0.998i)T \)
31 \( 1 + (0.846 + 0.532i)T \)
37 \( 1 + (-0.875 + 0.483i)T \)
41 \( 1 + (0.111 - 0.993i)T \)
43 \( 1 + (-0.745 + 0.666i)T \)
47 \( 1 + (0.960 - 0.276i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (0.985 + 0.167i)T \)
61 \( 1 + (0.993 - 0.111i)T \)
67 \( 1 + (-0.276 + 0.960i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.483 - 0.875i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.666 - 0.745i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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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

−30.1273461170174260368534179246, −28.35727531263745080294180299951, −26.85194099322563088859511452045, −26.37103798623879749608160460714, −25.67108394846139696431854134122, −24.64056402392965636689954364500, −23.37137096099076717102042497283, −22.44374689903441501681239554345, −21.40907716329211808649333156881, −20.59569090372900949566434578403, −19.164071237539381683689048042027, −17.78773457640559252532648684062, −16.95729995779999549679675069545, −15.54255540227274717196229684522, −14.87500894311376719973523715958, −13.74309358594876246662138043606, −13.32490701789955747349584407139, −11.17033919208728234860372033376, −9.899841503536323853159511785396, −8.85493748507165354002485094455, −7.40139000506637694025130477927, −6.65102756590123325707755582863, −4.80241921166314421005665000629, −3.88918024241447277140638188251, −2.38235861419567083576706348917, 1.003745180936521737208742954297, 2.22684278373682618283722057834, 3.40916715925117070530815882009, 5.143427900260217948877337427271, 6.19697138426659346522274556347, 8.60291269695154973356159811628, 8.7850203103898865179566439525, 10.34800887752660022428844922123, 11.84163786913037736417420634357, 12.84653316124482290427467946459, 13.49201475084398943532643216887, 14.594782166759198676127901923128, 15.76287782486722799829096423409, 17.6650493935941189098116263290, 18.53446186360968408027953313324, 19.44894450681556750569415935706, 20.633583199606375777664194151966, 21.1265482426566928914908634850, 22.25052862368877447345004869511, 23.71511690063554873346758904330, 24.626830248676297468258200580075, 25.24823885111926086313794028494, 26.83671456631137870555935422992, 27.89998876588948733205877681620, 29.01394670904819246456766437424

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