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Properties

Label 1-97-97.53-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.0323 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0323 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0323 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.0323 + 0.999i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.0323 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4614373061 + 0.4766225063i\)
\(L(\frac12)\) \(\approx\) \(0.4614373061 + 0.4766225063i\)
\(L(1)\) \(\approx\) \(0.6487542432 + 0.3053120264i\)
\(L(1)\) \(\approx\) \(0.6487542432 + 0.3053120264i\)

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

−29.5631598997222915434019958286, −28.91834150735897004118172433048, −27.84340605587969608595302697283, −26.710074700129330523117006686438, −25.49162046595830698354248833456, −24.72423998034911684726782267177, −24.00285095231204772036208184487, −22.7985929373367966164632251552, −21.07820280414344005487592892869, −19.95144923365157511041603326918, −19.2147028836097129219204102591, −17.82280067991895830488434429778, −17.12814154581935561898865631518, −16.58105437136122663421184972348, −14.58383285687392804068694077299, −13.60417489536549243137885285580, −12.228476513151331637153966049594, −11.08041878512123099867604506585, −9.69774582481677721343748473122, −8.66025298159638406555460694784, −7.1879095792979527017106510122, −6.54917861728920522958231954987, −4.957781221097274737833762174221, −2.214199150675510903451582890053, −1.00588401068206591089729343983, 2.12441963146979048156965313117, 3.46834117002951951302444062956, 5.55073294537092144652275444930, 6.61173987940528555337718425349, 8.53474920721225727999416452658, 9.41865876927208242207490252872, 10.4034131248049821104332576566, 11.33756386319971546330581628798, 12.57103212655754188942172972436, 14.72412294194406354889334828468, 15.25146948348228546519126102953, 16.898287015491594617277333072609, 17.35735519424574672034529639504, 18.62768458456008318710669062150, 19.68000458724089619293730044605, 21.08289528857547284978871993108, 21.73427557688192393400704346867, 22.51298290944128286876846481317, 24.58025308649197496451241450546, 25.46840405947393152769068282920, 26.33007634050484117284788553197, 27.44753920543484687504470060180, 28.01559854848618752869800883790, 29.19703288463631149804200856862, 29.93756162210761221582851025271

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