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Properties

Label 1-97-97.66-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.428 + 0.903i$
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  + (−0.258 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.428 + 0.903i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.428 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3260196028 + 0.5152770934i\)
\(L(\frac12)\) \(\approx\) \(0.3260196028 + 0.5152770934i\)
\(L(1)\) \(\approx\) \(0.5784224025 + 0.3785660953i\)
\(L(1)\) \(\approx\) \(0.5784224025 + 0.3785660953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 + (-0.608 + 0.793i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (-0.991 + 0.130i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.382 + 0.923i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (-0.130 - 0.991i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (0.130 + 0.991i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (-0.793 - 0.608i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.382 - 0.923i)T \)
71 \( 1 + (0.130 - 0.991i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.608 - 0.793i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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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

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

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