Properties

Degree 1
Conductor 97
Sign $0.162 + 0.986i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.162 + 0.986i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (93, \cdot )$
Sato-Tate  :  $\mu(24)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 97,\ (0:\ ),\ 0.162 + 0.986i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9233530376 + 0.7834215751i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9233530376 + 0.7834215751i\)
\(L(\chi,1)\)  \(\approx\)  \(1.129182391 + 0.5257091949i\)
\(L(1,\chi)\)  \(\approx\)  \(1.129182391 + 0.5257091949i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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