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Properties

Label 1-97-97.73-r0-0-0
Degree $1$
Conductor $97$
Sign $0.998 + 0.0618i$
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  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)6-s + (0.965 + 0.258i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.258 + 0.965i)10-s + (0.866 + 0.5i)11-s + i·12-s + (−0.258 + 0.965i)13-s + (0.965 − 0.258i)14-s + (−0.258 − 0.965i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)6-s + (0.965 + 0.258i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.258 + 0.965i)10-s + (0.866 + 0.5i)11-s + i·12-s + (−0.258 + 0.965i)13-s + (0.965 − 0.258i)14-s + (−0.258 − 0.965i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.998 + 0.0618i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.998 + 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.323130999 + 0.04096134150i\)
\(L(\frac12)\) \(\approx\) \(1.323130999 + 0.04096134150i\)
\(L(1)\) \(\approx\) \(1.334183667 + 0.02873652871i\)
\(L(1)\) \(\approx\) \(1.334183667 + 0.02873652871i\)

Euler product

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

−30.00726536178888099691208523268, −29.49530266843935179950592775397, −27.83622524811989583911946883582, −27.33388042193208847066117349870, −25.38258956133231966059043007487, −24.492155730731946873399860426, −23.91370565276186966306179034245, −23.00610643833294945160223470856, −21.87962222630637582008916452465, −20.85458172220987392036984791532, −19.710419196615254314655736734145, −17.97903789338762280109501320896, −16.91700720377399312805304925010, −16.44307189367477224979987727919, −14.91015092690432750908833731967, −13.7564475767801875412446344568, −12.4931626693288531428024204899, −11.95241422354973821900030391872, −10.69096648058004373460849465582, −8.37332572928868388208960643079, −7.569514808581362981024002519016, −5.99763621338241154305830206616, −5.12421578893916528900505974025, −3.95010433758386846609200956631, −1.52524377061595061123530148175, 1.91993624213367612101824925564, 3.77284822105727470878935383390, 4.74866344561766644539469087432, 6.10590617043631469115601324754, 7.17250565418747611371640030759, 9.51802846951397843761382329105, 10.680231900079635649658239060970, 11.60460239340091577889550458703, 12.16628385892786856644660078671, 14.16719340772100839582675064996, 14.79178759291275581273239513604, 15.86127808093224411263328627260, 17.33712838942996719624269266871, 18.48770594779765846018356643083, 19.62909807567348012418680137589, 21.13166454735386820553590113932, 21.72144115503457839142401150603, 22.68475118883251540973562207885, 23.50834438279288723497754517849, 24.45604839412290138694349718749, 26.03930101643811604022083548397, 27.53509585346602889613050489964, 27.9020590605788587952538301884, 29.2990032821720400144693892939, 30.15941388586722322167591894898

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