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Properties

Label 1-97-97.42-r1-0-0
Degree $1$
Conductor $97$
Sign $0.997 - 0.0700i$
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  + (0.382 − 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s + (−0.831 + 0.555i)5-s + i·6-s + (−0.831 − 0.555i)7-s + (−0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (0.195 + 0.980i)10-s + (0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s + (0.555 + 0.831i)13-s + (−0.831 + 0.555i)14-s + (0.555 − 0.831i)15-s + i·16-s + (0.555 + 0.831i)17-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)4-s + (−0.831 + 0.555i)5-s + i·6-s + (−0.831 − 0.555i)7-s + (−0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (0.195 + 0.980i)10-s + (0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s + (0.555 + 0.831i)13-s + (−0.831 + 0.555i)14-s + (0.555 − 0.831i)15-s + i·16-s + (0.555 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.997 - 0.0700i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.997 - 0.0700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9531229437 - 0.03343675582i\)
\(L(\frac12)\) \(\approx\) \(0.9531229437 - 0.03343675582i\)
\(L(1)\) \(\approx\) \(0.7563869754 - 0.2018155827i\)
\(L(1)\) \(\approx\) \(0.7563869754 - 0.2018155827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.382 - 0.923i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (-0.831 + 0.555i)T \)
7 \( 1 + (-0.831 - 0.555i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.555 + 0.831i)T \)
17 \( 1 + (0.555 + 0.831i)T \)
19 \( 1 + (0.831 - 0.555i)T \)
23 \( 1 + (-0.195 + 0.980i)T \)
29 \( 1 + (-0.195 + 0.980i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + T \)
67 \( 1 + (0.555 + 0.831i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.831 - 0.555i)T \)
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

−30.0697065487292978775531002020, −28.601545893530700063949420041284, −27.74789348341885736042827564756, −26.85558959040949750182836567219, −25.04746033760230004690693711490, −24.85222009352855639467163866292, −23.37312751941072619959185587994, −22.82363242404896316115633966635, −22.047619048412814775178895789155, −20.38185924590059323546472629686, −18.893772370905765012421375722049, −17.983092121660934477587765639785, −16.592607867346628396044526447228, −16.1567785479414509500866683514, −15.03005378268633961781281098658, −13.391615550755213917717366221009, −12.36239729427319311707054032003, −11.78135815859090915001348607808, −9.73270014028147531132651986697, −8.31112728769273086637410152188, −7.13234206581906972890670364295, −6.027583220090495578599377404573, −4.9422647608218568505378793511, −3.53336610319532152273335218507, −0.58502933804334681571469074389, 1.01164111898177683613676086546, 3.51397504024562439767538852046, 4.06967838661176604499745222327, 5.82647228892202196016117147277, 6.97411991013605309617582320562, 9.16220988330863437434883298616, 10.288039300552440482865732275041, 11.31955265648948440824695481626, 11.952727018272768260817171293708, 13.308622098717432949579827621213, 14.61924519636369646998383972242, 15.870241824671786658911861604928, 16.96222821248589390128656299999, 18.42852861217444875057724537369, 19.30185385694533792647635042878, 20.2766259754266332332294588214, 21.82121895952003823784157060198, 22.26264672451386806349815892023, 23.437311771715649743812333337170, 23.82500698981183180067693728133, 26.10866450636779853476076175466, 27.06265242038881081932063948319, 27.8874126790172828768387673568, 28.85995706034279995580099745650, 29.82023935946948062581414311429

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