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Properties

Label 1-97-97.24-r0-0-0
Degree $1$
Conductor $97$
Sign $0.162 - 0.986i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9233530376 - 0.7834215751i\)
\(L(\frac12)\) \(\approx\) \(0.9233530376 - 0.7834215751i\)
\(L(1)\) \(\approx\) \(1.129182391 - 0.5257091949i\)
\(L(1)\) \(\approx\) \(1.129182391 - 0.5257091949i\)

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

−30.39914178673521642445068112220, −29.3533532249528102785996757261, −28.79660765519057893041856221608, −26.969292679852080946004723625614, −25.9539966874058415338936643863, −24.91774331367835424003593334035, −23.98051992731406721140901406481, −22.791129840658936537258084740849, −22.27104773797446729350463331894, −21.51947045690055151294685650793, −19.58664868255756309604592485920, −18.54188255632608130513529691700, −17.3183581522920113251009112217, −16.361877353887219607623318752333, −15.30611926542831178676958784776, −13.864264050944878979849294543546, −13.17013567905889278655122835462, −11.72540945103763265934881587799, −11.09601383417335008001299521543, −9.229254831928523875705024011869, −7.14483369403804040614870451612, −6.59003226109699127535906424316, −5.64125833595277105287801607517, −3.91765588614146832932871044304, −2.39900547152328344605621137642, 1.19009174757681639501543612445, 3.47065681359608906518986907090, 4.620942139143793622655675386767, 5.72056480849911468816644914231, 6.76518028817863615242832863725, 9.2505584406570011730505482127, 10.124424552329057514329409559503, 11.35051865002488989623257023697, 12.60608056606254057616080645896, 13.06485300015494941159715217818, 14.779743997804264753667251560108, 15.965844174128920891147438611114, 16.72387690150049655728754904022, 18.064019958664161493943009749461, 19.82489877679256820275597756126, 20.406682751793910191592748453333, 21.649479571285315669072926799325, 22.541915082826809506207798950603, 23.20775333629633388078584356848, 24.44018548503057813012965354882, 25.38208695264363023446117039244, 27.19530409569464581753791625308, 28.119972291634409868257021369730, 28.97619319951211232767896430219, 29.54934517180925825590751233702

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