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Properties

Label 1-97-97.44-r0-0-0
Degree $1$
Conductor $97$
Sign $0.752 - 0.659i$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
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  + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.5i)4-s + (−0.793 − 0.608i)5-s + (0.5 − 0.866i)6-s + (−0.130 − 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (0.608 + 0.793i)10-s + (0.965 − 0.258i)11-s + (−0.707 + 0.707i)12-s + (0.793 + 0.608i)13-s + (−0.130 + 0.991i)14-s + (0.793 − 0.608i)15-s + (0.5 + 0.866i)16-s + (0.130 − 0.991i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.5i)4-s + (−0.793 − 0.608i)5-s + (0.5 − 0.866i)6-s + (−0.130 − 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (0.608 + 0.793i)10-s + (0.965 − 0.258i)11-s + (−0.707 + 0.707i)12-s + (0.793 + 0.608i)13-s + (−0.130 + 0.991i)14-s + (0.793 − 0.608i)15-s + (0.5 + 0.866i)16-s + (0.130 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.752 - 0.659i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.752 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4871141253 - 0.1832337025i\)
\(L(\frac12)\) \(\approx\) \(0.4871141253 - 0.1832337025i\)
\(L(1)\) \(\approx\) \(0.5886509276 - 0.06894068786i\)
\(L(1)\) \(\approx\) \(0.5886509276 - 0.06894068786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.793 - 0.608i)T \)
7 \( 1 + (-0.130 - 0.991i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (0.793 + 0.608i)T \)
17 \( 1 + (0.130 - 0.991i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.991 - 0.130i)T \)
29 \( 1 + (0.608 - 0.793i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (-0.608 + 0.793i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (-0.991 + 0.130i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.130 + 0.991i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
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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

−30.31323359390310242720754767055, −28.961337626690533461773740714071, −28.071545846776564897730240528566, −27.30970254358341289899919785351, −25.84061546315588377608907151233, −25.1863044823094041988889413364, −24.148741593202966195552537035562, −23.15042783766862697907059906830, −22.03033031220768683618552387667, −20.08919883980148008345894833583, −19.3830570705978771732139776172, −18.40057678769673547803160169927, −17.825506663702737965599330325212, −16.391927575674338835371005964404, −15.32194713452269140341530300525, −14.25417753532892602683615756571, −12.292257459057665928969464088624, −11.67555845537858181076853085742, −10.4388857569902158143027434928, −8.73914052014954801825308902124, −7.89669226863117306164771482849, −6.67587754754885714084233682410, −5.80568453384714013523325774963, −3.130875451410992192507322484399, −1.51642478421911680689078598984, 0.84834384507078265871389148731, 3.433916326161459415015497048459, 4.38337801803883289486101504740, 6.40990570808993046255135830428, 7.85429261656679930447077589120, 9.07020641800698001538738773039, 9.93492918423759656278925147617, 11.37957511600404641779336750825, 11.771076619416810773040326500961, 13.75595966863652001113224029303, 15.4048386482211080802677833957, 16.46498902343426525587230520053, 16.74151748570648242236605476121, 18.20371393422120552723663440944, 19.762269306022314566822792176363, 20.21476467151295100394462661872, 21.225754462455383366297320077087, 22.565706416590939225132613315445, 23.71589632593173866586954215806, 24.970266790581698585505364221885, 26.428624774348855949784335575789, 26.863903587847898466804508250799, 27.875649733536306756394133715410, 28.53808803576495485321443674718, 29.733655144225569611589199471066

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