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Properties

Label 1-647-647.306-r0-0-0
Degree $1$
Conductor $647$
Sign $0.627 - 0.778i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (−0.602 + 0.798i)2-s + (0.0922 + 0.995i)3-s + (−0.273 − 0.961i)4-s + (−0.982 + 0.183i)5-s + (−0.850 − 0.526i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s + (0.445 − 0.895i)10-s + (−0.982 − 0.183i)11-s + (0.932 − 0.361i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.273 − 0.961i)15-s + (−0.850 + 0.526i)16-s + (0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (0.0922 + 0.995i)3-s + (−0.273 − 0.961i)4-s + (−0.982 + 0.183i)5-s + (−0.850 − 0.526i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s + (0.445 − 0.895i)10-s + (−0.982 − 0.183i)11-s + (0.932 − 0.361i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.273 − 0.961i)15-s + (−0.850 + 0.526i)16-s + (0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (306, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ 0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2011022825 - 0.09624714499i\)
\(L(\frac12)\) \(\approx\) \(0.2011022825 - 0.09624714499i\)
\(L(1)\) \(\approx\) \(0.4805882140 + 0.2779852748i\)
\(L(1)\) \(\approx\) \(0.4805882140 + 0.2779852748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (-0.602 + 0.798i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (0.932 - 0.361i)T \)
17 \( 1 + (0.0922 + 0.995i)T \)
19 \( 1 + (-0.850 + 0.526i)T \)
23 \( 1 + (-0.850 - 0.526i)T \)
29 \( 1 + (-0.850 - 0.526i)T \)
31 \( 1 + (-0.982 - 0.183i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (0.932 + 0.361i)T \)
43 \( 1 + (-0.602 - 0.798i)T \)
47 \( 1 + (0.0922 + 0.995i)T \)
53 \( 1 + (-0.850 - 0.526i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.0922 - 0.995i)T \)
67 \( 1 + (0.0922 - 0.995i)T \)
71 \( 1 + (-0.850 - 0.526i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.602 - 0.798i)T \)
83 \( 1 + (0.445 + 0.895i)T \)
89 \( 1 + (-0.602 + 0.798i)T \)
97 \( 1 + (0.445 - 0.895i)T \)
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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

−23.212379964669392834022343309, −22.08750036373031894644362112168, −20.94038435013281340411951782062, −20.46957699744412342866571321831, −19.65089367114282896136556834732, −18.74870384625651624901101102591, −18.3003755810735663307751984302, −17.66104629101439666985737301327, −16.5218549017020345897672985487, −15.68573180729955001386333775558, −14.536999586949553995718101332420, −13.45530666941106485666963002481, −12.79084911487926617378118554843, −11.83007875345302298606333578756, −11.39354553193856046071668521131, −10.61003929176894027529268276783, −9.00340610281725096486131089836, −8.50303038238044339258049865096, −7.66454086182827825270609072593, −7.11247294885930944260805313377, −5.48905905633052723266164800658, −4.38303472756382984352877102500, −3.23032588238844658911273468998, −2.20477216453136709418266945888, −1.28307660279905394556487932500, 0.14026722407759909558063270153, 1.939427583507030440418174957827, 3.65253789848697100959137501693, 4.32834987030338746817528683761, 5.35207886127401428596889954592, 6.20967301978614929416341114828, 7.710657679837124746278847764446, 8.13823085851093647145063598851, 8.77725428467788469305591592453, 10.155054490481433007733098586040, 10.823442320916250922550791726387, 11.19688577806843192056034379924, 12.780960056239785342899456718130, 14.06981074473078642048785408280, 14.74618608682224300867514182228, 15.40325428226610123954515558317, 16.05886694724067589369691962885, 16.79253967135764300408192769592, 17.69447238250622855377367985553, 18.57781986982114028926750477831, 19.37009420336870119210842528541, 20.42167551472810052388904596289, 20.83505551529114641241073060341, 22.05912280749989476990858318587, 23.070273459520527878421569555158

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