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Properties

Label 1-697-697.526-r1-0-0
Degree $1$
Conductor $697$
Sign $0.0964 - 0.995i$
Analytic cond. $74.9030$
Root an. cond. $74.9030$
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  + (−0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (0.987 − 0.156i)6-s + (0.987 + 0.156i)7-s + (0.951 + 0.309i)8-s + i·9-s + (−0.309 + 0.951i)10-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (−0.156 − 0.987i)13-s + (−0.707 + 0.707i)14-s + (−0.891 − 0.453i)15-s + (−0.809 + 0.587i)16-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (0.987 − 0.156i)6-s + (0.987 + 0.156i)7-s + (0.951 + 0.309i)8-s + i·9-s + (−0.309 + 0.951i)10-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (−0.156 − 0.987i)13-s + (−0.707 + 0.707i)14-s + (−0.891 − 0.453i)15-s + (−0.809 + 0.587i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(697\)    =    \(17 \cdot 41\)
Sign: $0.0964 - 0.995i$
Analytic conductor: \(74.9030\)
Root analytic conductor: \(74.9030\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{697} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 697,\ (1:\ ),\ 0.0964 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047944937 - 0.9513058624i\)
\(L(\frac12)\) \(\approx\) \(1.047944937 - 0.9513058624i\)
\(L(1)\) \(\approx\) \(0.8536862802 - 0.1037687794i\)
\(L(1)\) \(\approx\) \(0.8536862802 - 0.1037687794i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.987 + 0.156i)T \)
11 \( 1 + (0.891 - 0.453i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.453 - 0.891i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.987 + 0.156i)T \)
53 \( 1 + (0.453 - 0.891i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.987 - 0.156i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−22.12341318687200762266029191655, −21.7667723860414171398181158115, −21.181295287084713608122538984351, −20.34149745991027842735952900561, −19.50902833467921553891125259149, −18.20027484772737644737851974995, −17.82305618275962808290374657237, −17.09665436666017352352924531035, −16.5596502249742032687658333043, −15.26029703726917167746484856586, −14.26039790089996029480838276381, −13.583860440509818154224014881487, −12.158802035508383795312039450864, −11.69923416538936366916977257917, −10.79832495266351030062023906486, −10.17487252059944286484258840254, −9.28176552575557812204894394743, −8.73685272721452310040015451839, −7.188108249719946291180577436165, −6.44887471690772540557471993115, −5.007220123558488649287795988365, −4.44074419631857293932066617921, −3.281236183290891924795964212689, −1.92034327991521747936469223712, −1.21325371117747974804395818512, 0.48040246191624819363752568037, 1.40305086503985318844860491212, 2.198135994868909534274988014749, 4.3922610324419750990134562080, 5.391516744539642030213893689734, 5.95840399388033620536132072964, 6.66350669340137119290595579493, 8.019337840706163981519232843984, 8.267242502632821606619707738665, 9.59246563447404624172724637678, 10.39344967793771213133872319609, 11.27874354439854015072942850674, 12.27407877461704371909619527808, 13.29403045234005846888161698034, 14.13792222515150744504453013009, 14.69212344814224743302693556587, 15.990865561929803615430666772493, 16.866463726253224457079127962051, 17.34792046757844578393338916381, 17.99421213881113051831484158260, 18.59992546956142806628947954793, 19.58831610579925428957504602702, 20.53926948707211574945122286344, 21.62401682843854599462497840312, 22.50272891089071926021551044228

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