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Properties

Label 1-1700-1700.1483-r0-0-0
Degree $1$
Conductor $1700$
Sign $-0.400 + 0.916i$
Analytic cond. $7.89476$
Root an. cond. $7.89476$
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  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.587 + 0.809i)11-s + (0.587 − 0.809i)13-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.951 + 0.309i)29-s + (−0.951 + 0.309i)31-s + (−0.951 + 0.309i)33-s + (0.809 + 0.587i)37-s + (0.587 + 0.809i)39-s + (0.587 − 0.809i)41-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.587 + 0.809i)11-s + (0.587 − 0.809i)13-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.951 + 0.309i)29-s + (−0.951 + 0.309i)31-s + (−0.951 + 0.309i)33-s + (0.809 + 0.587i)37-s + (0.587 + 0.809i)39-s + (0.587 − 0.809i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.400 + 0.916i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ -0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5277714601 + 0.8062410247i\)
\(L(\frac12)\) \(\approx\) \(0.5277714601 + 0.8062410247i\)
\(L(1)\) \(\approx\) \(0.7816424790 + 0.3056687227i\)
\(L(1)\) \(\approx\) \(0.7816424790 + 0.3056687227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−19.825138873233011888661592265478, −19.26449274090803824311803037523, −18.66156035986173348163406698406, −18.09971448012830159899970509867, −16.98847369552191340571391997119, −16.48173868139703741151999380134, −15.977199741877566118273820397341, −14.64406670889991897722750998755, −13.95632124744096002393064521853, −13.39734919377175376926089416810, −12.506484672367041335101215556819, −11.99716495157664542864593094972, −11.14832014280797215828138097081, −10.38127691853272295861040597238, −9.30565795635541385800380779133, −8.61932414050737292984009093445, −7.781701939205178834394399802643, −6.82740959126941995533035514253, −6.12453210221439996787188532921, −5.828925687420782199652117164267, −4.308446945941141431690924203261, −3.511493681370436592723906483713, −2.484474882433385713104637279127, −1.53856484584931261190631074337, −0.44614875644587987638647116073, 0.96154207787641037583288489437, 2.49091775990808348548938798752, 3.360362892114364883135971891926, 4.078383195182941889504536740241, 4.919201169909633747666481742021, 5.90192639640033909386478782743, 6.47687648252087121404238605033, 7.45543292363254265460551814033, 8.618403136479526012811492808985, 9.3223401209046672370981053360, 9.93126452397073537149837171489, 10.64507157884427254836116326717, 11.42192734716910085752720043636, 12.33133519652050514292080522710, 12.9532342450965342347426653957, 13.93923553812098231560793892046, 14.78330766323612713081048300433, 15.59320652655825302378965047954, 15.934281585692468558947584868380, 16.79736876573565677087352339889, 17.58587187693948467845879169501, 18.0833006902148632292812846424, 19.325095611929741817892752384506, 19.990964981908392542283104952417, 20.393526491065592133626997807

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