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Properties

Label 1-1375-1375.1116-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.0708 + 0.997i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.968 + 0.248i)2-s + (−0.425 − 0.904i)3-s + (0.876 + 0.481i)4-s + (−0.187 − 0.982i)6-s + (0.309 + 0.951i)7-s + (0.728 + 0.684i)8-s + (−0.637 + 0.770i)9-s + (0.0627 − 0.998i)12-s + (−0.637 + 0.770i)13-s + (0.0627 + 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.425 + 0.904i)17-s + (−0.809 + 0.587i)18-s + (−0.187 − 0.982i)19-s + (0.728 − 0.684i)21-s + ⋯
L(s)  = 1  + (0.968 + 0.248i)2-s + (−0.425 − 0.904i)3-s + (0.876 + 0.481i)4-s + (−0.187 − 0.982i)6-s + (0.309 + 0.951i)7-s + (0.728 + 0.684i)8-s + (−0.637 + 0.770i)9-s + (0.0627 − 0.998i)12-s + (−0.637 + 0.770i)13-s + (0.0627 + 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.425 + 0.904i)17-s + (−0.809 + 0.587i)18-s + (−0.187 − 0.982i)19-s + (0.728 − 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.0708 + 0.997i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.0708 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.514147729 + 1.410438864i\)
\(L(\frac12)\) \(\approx\) \(1.514147729 + 1.410438864i\)
\(L(1)\) \(\approx\) \(1.507872455 + 0.3343643335i\)
\(L(1)\) \(\approx\) \(1.507872455 + 0.3343643335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.968 + 0.248i)T \)
3 \( 1 + (-0.425 - 0.904i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.637 + 0.770i)T \)
17 \( 1 + (-0.425 + 0.904i)T \)
19 \( 1 + (-0.187 - 0.982i)T \)
23 \( 1 + (0.535 - 0.844i)T \)
29 \( 1 + (-0.992 + 0.125i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (-0.637 + 0.770i)T \)
41 \( 1 + (-0.637 + 0.770i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.187 + 0.982i)T \)
53 \( 1 + (-0.187 + 0.982i)T \)
59 \( 1 + (-0.929 - 0.368i)T \)
61 \( 1 + (-0.929 + 0.368i)T \)
67 \( 1 + (-0.425 + 0.904i)T \)
71 \( 1 + (-0.425 - 0.904i)T \)
73 \( 1 + (0.968 + 0.248i)T \)
79 \( 1 + (0.728 - 0.684i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (0.0627 + 0.998i)T \)
97 \( 1 + (0.876 + 0.481i)T \)
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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

−20.91270951369303774731597936081, −20.14483831203284926502750878627, −19.59679282995302874844471981720, −18.381677905855269698644522865816, −17.286474053273587143188873772076, −16.83001814401161478403803810851, −15.94404249583203394124634750841, −15.24436516835227514545486253016, −14.59427812127050809624049295946, −13.838216252637323297403729989377, −13.052918590193644629428797414694, −12.0852621889140905609759444417, −11.400070533203224928175297092477, −10.69474005801346673674302041108, −10.0769678695830841073354356534, −9.32697810552330531807067582531, −7.870457566179635381488686898515, −7.1340172634718926568501634578, −6.11536314257611733445716681671, −5.250011227489863118021804489138, −4.69834912861954717807881754124, −3.78301321970759640654087019486, −3.181149599230218582605673583594, −1.923414509075539752197793111910, −0.55764228281119242009779843366, 1.5530370918212006549314000651, 2.28194622250228575498197273475, 3.06615278850146784175186465247, 4.5711317379843931071583729855, 5.02550181243447927249280761412, 6.08706810375744074282639690123, 6.59167433223095968284512107364, 7.402662055850549179243525689757, 8.331614743221988329340756853854, 9.05085867205105951332306831835, 10.637654562845364688964715656275, 11.26177794758917539884240597231, 12.09693684939196616084113418461, 12.48736132112781161673990052243, 13.33178176564245171127291556262, 14.04850851647647064847878978187, 14.92308941042546379940191005460, 15.44673949895318746478125266345, 16.569491399691531376465837239321, 17.14674468364704263039774989852, 17.86749400028847926276566376025, 18.91135006850753381280533148298, 19.39334126546214982368833453486, 20.34307917995168762947182516781, 21.28898610819042897603890287401

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