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Properties

Label 1-97-97.89-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.999 + 0.0332i$
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  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (−0.382 − 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (0.923 − 0.382i)15-s − 16-s + (0.923 + 0.382i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (−0.382 − 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (0.923 − 0.382i)15-s − 16-s + (0.923 + 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.999 + 0.0332i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.999 + 0.0332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01147864501 + 0.6896174230i\)
\(L(\frac12)\) \(\approx\) \(0.01147864501 + 0.6896174230i\)
\(L(1)\) \(\approx\) \(0.5619089973 + 0.6236688536i\)
\(L(1)\) \(\approx\) \(0.5619089973 + 0.6236688536i\)

Euler product

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

−29.90010582781692370672477058362, −28.697703983075517933489804523212, −27.9513886970441895662398598355, −26.680361476778250548168062437977, −25.14486638994692398306932809689, −23.80674156449074773245749473684, −23.1252599020562828554822212290, −22.66105622458560607880833010364, −21.26571612376548565977493416222, −19.93194206450467871810847845546, −18.92345329209716775492679226901, −18.46157038067207337214177391742, −16.50280520997351389197614296030, −15.63237939307459377232526190953, −14.04620489709315891930772529282, −13.009090948986143067306060061160, −12.18193020526541552014049652212, −11.00802328151735551886392710949, −10.30745145303655778352894198672, −8.13114796845597112555922807151, −6.668283328060704666763877019745, −5.70847717789904996400793306831, −4.02271608482538590295681293327, −2.78145833969470577747690793465, −0.61470893834242409446203382910, 3.34184514845998857394915016569, 4.3455521713221917493802777037, 5.558776100751779280780473532672, 6.69079768058045742877559198434, 8.15567312024653478815079378386, 9.50009442799559820738050519711, 11.1426613010854796964528914482, 12.26986105456210305556318217392, 13.02668992810146227480077237335, 14.8846907111863502715327417760, 15.73867264818550497826888392092, 16.26349271032566615620016668505, 17.3919015434418162971473258146, 18.842917787858720940820085337285, 20.46497354832516562441609912317, 21.3771750195312373219305090069, 22.50072880251996131735867725685, 23.36679566726832714614594394955, 23.82620941693041473785200948509, 25.63277092542418720573751695532, 26.15670336294888645735316143847, 27.622101372469262019769810036157, 28.302990268340011366893386732557, 29.61161040860200355430708260646, 30.99783970156270141783762290333

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