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Properties

Label 1-97-97.57-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.459 + 0.888i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
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  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.896 − 0.442i)5-s + (−0.866 − 0.5i)6-s + (0.0654 − 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (0.946 + 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (−0.442 + 0.896i)13-s + (0.0654 + 0.997i)14-s + (−0.442 − 0.896i)15-s + (0.866 − 0.5i)16-s + (0.997 − 0.0654i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.896 − 0.442i)5-s + (−0.866 − 0.5i)6-s + (0.0654 − 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (0.946 + 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (−0.442 + 0.896i)13-s + (0.0654 + 0.997i)14-s + (−0.442 − 0.896i)15-s + (0.866 − 0.5i)16-s + (0.997 − 0.0654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.459 + 0.888i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4477187226 + 0.7359265030i\)
\(L(\frac12)\) \(\approx\) \(0.4477187226 + 0.7359265030i\)
\(L(1)\) \(\approx\) \(0.6898184951 + 0.2586465909i\)
\(L(1)\) \(\approx\) \(0.6898184951 + 0.2586465909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.991 + 0.130i)T \)
3 \( 1 + (0.793 + 0.608i)T \)
5 \( 1 + (-0.896 - 0.442i)T \)
7 \( 1 + (0.0654 - 0.997i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (-0.442 + 0.896i)T \)
17 \( 1 + (0.997 - 0.0654i)T \)
19 \( 1 + (-0.831 + 0.555i)T \)
23 \( 1 + (0.751 + 0.659i)T \)
29 \( 1 + (-0.946 + 0.321i)T \)
31 \( 1 + (-0.608 + 0.793i)T \)
37 \( 1 + (0.659 + 0.751i)T \)
41 \( 1 + (-0.321 - 0.946i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.130 - 0.991i)T \)
59 \( 1 + (-0.751 + 0.659i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (-0.321 + 0.946i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.0654 - 0.997i)T \)
89 \( 1 + (0.923 - 0.382i)T \)
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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

−29.712735901455658318832071941896, −28.28624807220856669134363704457, −27.29347395964114249229083274385, −26.47461517777750529740441577333, −25.40823683957672614408029053505, −24.61516075411811822339236750936, −23.598364383339373641508499424228, −21.88980568856898216400009388697, −20.70558443660403675657809542959, −19.59129142199283846069912769848, −18.841223855870429499528326427268, −18.30871708177448618774205128888, −16.73966513901711523057862015727, −15.30775141940637155514688400937, −14.82712980276241254022903191350, −12.856721978913677830595794381419, −11.89706730728402458710266710582, −10.735977674749915285979968792091, −9.15584059957926479443538961524, −8.23495142541214851069517379008, −7.443147712578242248945348189485, −6.0285830201773741745392718154, −3.35086803085185886905623050902, −2.44503484594241922902741717513, −0.48147420671365513060644592707, 1.61501648512957763488704388499, 3.49958683067354996898912971989, 4.785301105270811693264132643422, 7.18482934291289894255120846007, 7.82523592030446442086986038464, 9.09899881181905421607572953082, 10.05916278197457491501537316771, 11.17501364456038024980017617687, 12.60381898172051454298626929795, 14.40422438058613271191460761908, 15.228494551932313259164154680061, 16.443817108991170949274320951312, 17.01199285114731061900600380614, 18.80626022912423724849927347315, 19.64091834864143813080580235076, 20.41997212415486328522324624200, 21.19911041567974527014512534121, 23.15251359734032971348850479949, 24.05981229659638316297086651720, 25.384970462292823434987364083647, 26.121775057635126688692457571944, 27.2705001845752173126506070801, 27.55576496323306629633304326088, 28.88495695233967002075626507618, 30.1897346003795122076723842821

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