Properties

Label 1-97-97.7-r1-0-0
Degree $1$
Conductor $97$
Sign $0.903 - 0.428i$
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 + (−0.442 + 0.896i)5-s + (−0.866 − 0.5i)6-s + (0.997 + 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.321 + 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (0.896 + 0.442i)13-s + (0.997 − 0.0654i)14-s + (0.896 − 0.442i)15-s + (0.866 − 0.5i)16-s + (−0.0654 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (−0.793 − 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.442 + 0.896i)5-s + (−0.866 − 0.5i)6-s + (0.997 + 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.321 + 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (0.896 + 0.442i)13-s + (0.997 − 0.0654i)14-s + (0.896 − 0.442i)15-s + (0.866 − 0.5i)16-s + (−0.0654 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.903 - 0.428i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.903 - 0.428i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.660173935 - 0.5983128454i\)
\(L(\frac12)\) \(\approx\) \(2.660173935 - 0.5983128454i\)
\(L(1)\) \(\approx\) \(1.715301399 - 0.2754629874i\)
\(L(1)\) \(\approx\) \(1.715301399 - 0.2754629874i\)

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 \( 1 + (-0.442 + 0.896i)T \)
7 \( 1 + (0.997 + 0.0654i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (0.896 + 0.442i)T \)
17 \( 1 + (-0.0654 - 0.997i)T \)
19 \( 1 + (0.555 + 0.831i)T \)
23 \( 1 + (0.659 - 0.751i)T \)
29 \( 1 + (0.321 + 0.946i)T \)
31 \( 1 + (0.608 - 0.793i)T \)
37 \( 1 + (-0.751 + 0.659i)T \)
41 \( 1 + (-0.946 + 0.321i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (-0.659 - 0.751i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (-0.946 - 0.321i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.997 + 0.0654i)T \)
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

−30.23906846739486605228726152048, −28.586077076034468925490328622491, −28.18499647836333601834227860886, −26.98738639715534558334909942312, −25.50474018372996068054867379082, −24.28971904887403411617525520751, −23.52322258756008650544264480991, −22.77726382266426104906192259160, −21.413260869671714270255454111352, −20.806819555134418660500334056054, −19.846239460346781793254296117897, −17.6681858733020431817443245051, −16.98169976913120296348969367250, −15.573089462510616790764427305156, −15.21477781399986511428058621731, −13.52685530038969535189854861658, −12.30081342524404502734421375787, −11.53909899054440834353069446562, −10.45188999050064176010284843727, −8.61982553817530700624103926747, −7.175031034133373743303887798216, −5.578036636907507329604124530868, −4.7496901837817633343815208396, −3.801856366683497172314564596053, −1.38573551985341876563404576205, 1.31726460654838948876079779794, 3.00919285745367681326631864350, 4.59985457205475280426122116507, 5.90078101423941759712343368812, 6.89662639081944629705501919407, 8.06400208031701170794394404714, 10.64880752539872860225455218614, 11.34052822391748615873870075301, 12.02582346469062630966600178031, 13.64375079449274316430827033954, 14.29513336463265940040601807551, 15.70905473124973050936246948630, 16.72294624537224530748863473480, 18.36525405940407583065557566330, 18.9192388697296074896440464902, 20.501429905336788263193456363547, 21.644315792306490156586259511, 22.582593144700852957803019486926, 23.39517846253035123551070901086, 24.2411962555999811599228220307, 25.129689288553543223186274242884, 26.78007798361434334050728786016, 27.87297335396622333064351664010, 29.153539821032255873102116683693, 29.868930839519437099822351863943

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