Properties

Degree 1
Conductor 97
Sign $-0.428 - 0.903i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.258 − 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.991 + 0.130i)5-s + (0.5 + 0.866i)6-s + (−0.608 − 0.793i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 − 0.991i)10-s + (0.258 − 0.965i)11-s + (0.707 − 0.707i)12-s + (−0.991 − 0.130i)13-s + (−0.608 + 0.793i)14-s + (−0.991 + 0.130i)15-s + (0.5 − 0.866i)16-s + (0.608 − 0.793i)17-s + ⋯
L(s,χ)  = 1  + (−0.258 − 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.991 + 0.130i)5-s + (0.5 + 0.866i)6-s + (−0.608 − 0.793i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 − 0.991i)10-s + (0.258 − 0.965i)11-s + (0.707 − 0.707i)12-s + (−0.991 − 0.130i)13-s + (−0.608 + 0.793i)14-s + (−0.991 + 0.130i)15-s + (0.5 − 0.866i)16-s + (0.608 − 0.793i)17-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.428 - 0.903i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (25, \cdot )$
Sato-Tate  :  $\mu(48)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (0:\ ),\ -0.428 - 0.903i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3260196028 - 0.5152770934i$
$L(\frac12,\chi)$  $\approx$  $0.3260196028 - 0.5152770934i$
$L(\chi,1)$  $\approx$  0.5784224025 - 0.3785660953i
$L(1,\chi)$  $\approx$  0.5784224025 - 0.3785660953i

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

−30.291365234525330660022813231968, −29.03629016294658287606710659250, −28.34847457690223201113954461319, −27.4719124411501786431301343510, −25.968185713295160847212605524026, −25.14320180343274881587389329421, −24.39788389322992051593739908313, −23.12727786879181044106120598853, −22.28923305758999429797174858503, −21.44810473896669137738260122943, −19.37222707000765663681812701067, −18.39298632314140838716096381136, −17.38335093688456758714442086054, −16.82940516450530550143244470545, −15.56438995059966594834909045179, −14.40616784674800056636137552545, −12.9716528785021434255952993080, −12.18921185825655768463443034566, −10.06570859914647751523203577890, −9.631544447719641632583443926137, −7.850151071807481341928628523240, −6.4260280818012673350649825092, −5.82536336296575339377616570849, −4.610210870694716803684423449258, −1.800556510017643046127329854729, 0.81598464559817773464736156998, 2.793810380948604369346714285195, 4.3778734822740506626845673739, 5.711785535148808007890837047633, 7.13695601739221471669249543481, 9.18922895993196199079189077199, 10.10496533123669920070759025065, 10.85462804834119853069142237669, 12.137938673390055030746754644168, 13.21934625358975405714739228507, 14.23224903566671241536215950479, 16.37807090458784632541354790107, 17.07668603075826482198191921154, 17.994742723353564095880898953491, 19.11476901284578537694483053479, 20.35865428174402450259840912475, 21.56060096284452648669525470341, 22.13567443686250234243762387359, 23.06796591866266992074952366453, 24.41679909908162343645369474693, 26.0669957509785809303347050274, 26.83512397507226316501323913221, 27.87945993838658280175204825670, 29.005032966576110049373233691594, 29.597083841906887331656569568604

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