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Properties

Label 1-1339-1339.107-r0-0-0
Degree $1$
Conductor $1339$
Sign $0.847 + 0.531i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (−0.273 + 0.961i)2-s + (0.992 − 0.122i)3-s + (−0.850 − 0.526i)4-s + (0.650 − 0.759i)5-s + (−0.153 + 0.988i)6-s + (0.739 + 0.673i)7-s + (0.739 − 0.673i)8-s + (0.969 − 0.243i)9-s + (0.552 + 0.833i)10-s + (0.969 − 0.243i)11-s + (−0.908 − 0.417i)12-s + (−0.850 + 0.526i)14-s + (0.552 − 0.833i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + (−0.0307 + 0.999i)18-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (0.992 − 0.122i)3-s + (−0.850 − 0.526i)4-s + (0.650 − 0.759i)5-s + (−0.153 + 0.988i)6-s + (0.739 + 0.673i)7-s + (0.739 − 0.673i)8-s + (0.969 − 0.243i)9-s + (0.552 + 0.833i)10-s + (0.969 − 0.243i)11-s + (−0.908 − 0.417i)12-s + (−0.850 + 0.526i)14-s + (0.552 − 0.833i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + (−0.0307 + 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $0.847 + 0.531i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ 0.847 + 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.446226843 + 0.7035906530i\)
\(L(\frac12)\) \(\approx\) \(2.446226843 + 0.7035906530i\)
\(L(1)\) \(\approx\) \(1.544454930 + 0.4211872772i\)
\(L(1)\) \(\approx\) \(1.544454930 + 0.4211872772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.273 + 0.961i)T \)
3 \( 1 + (0.992 - 0.122i)T \)
5 \( 1 + (0.650 - 0.759i)T \)
7 \( 1 + (0.739 + 0.673i)T \)
11 \( 1 + (0.969 - 0.243i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
19 \( 1 + (-0.389 - 0.920i)T \)
23 \( 1 + (0.969 + 0.243i)T \)
29 \( 1 + (-0.982 - 0.183i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (-0.908 - 0.417i)T \)
41 \( 1 + (-0.982 + 0.183i)T \)
43 \( 1 + (0.816 + 0.577i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.389 - 0.920i)T \)
59 \( 1 + (0.213 + 0.976i)T \)
61 \( 1 + (-0.153 - 0.988i)T \)
67 \( 1 + (-0.952 - 0.303i)T \)
71 \( 1 + (-0.982 + 0.183i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.982 - 0.183i)T \)
83 \( 1 + (0.213 - 0.976i)T \)
89 \( 1 + (-0.0307 - 0.999i)T \)
97 \( 1 + (-0.153 + 0.988i)T \)
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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

−20.7131773326939934302603607285, −20.35231886100040075715672604045, −19.11957738346269266101505433437, −18.921559399478518449718307995649, −18.04001347398224138646147378303, −17.12170839557107958574215827429, −16.683567959942054111124721947030, −14.998644798525360415864620910294, −14.567751412499418182530693084098, −13.89376618786706526463397816832, −13.348223188807509288058553350336, −12.31190976529255837852717514912, −11.431388345042493313987462864, −10.51826127150998956163778969547, −10.03694850956341703402654763076, −9.27716188289183597391352529430, −8.481224323016473447784148398791, −7.55877104769378433478269128449, −6.946895150592293925175631000971, −5.47209040929723412796684564258, −4.36168351912472937846528031476, −3.65504417768968335003549795079, −2.8980154569108810006864382622, −1.817396085853696019688111328865, −1.345355191758576374291255385540, 1.20835137437884309868039861831, 1.73366605750050966162936541029, 3.12830628645637934587241602690, 4.31885916461377264802603868516, 5.03625392876646481378292250871, 5.90775099537834660769820140018, 6.81201016425637052873595677980, 7.74282237147088692399680481253, 8.578250059224628009869627941498, 8.99446932301945229622920270515, 9.55947654129121333437462574696, 10.59662961779028268478544940466, 11.9099179213210279964398686205, 12.860352531612844024023134335386, 13.45565862573245475688633775205, 14.434336582801114485432189399041, 14.64893067963533958209600011782, 15.59375705321015345295904732608, 16.36212160747668702936900603936, 17.36149498176844076969661539718, 17.64239330419566868271635365546, 18.80563337568471609752557603095, 19.23032409104599115394361303395, 20.115227492605193944547415203149, 21.15423967186983530085508054910

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