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Properties

Label 1-97-97.25-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: $\chi_{97} (25, \cdot )$
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

−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