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Properties

Label 1-4235-4235.412-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.988 + 0.150i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.717 + 0.696i)2-s + (−0.951 + 0.309i)3-s + (0.0285 − 0.999i)4-s + (0.466 − 0.884i)6-s + (0.676 + 0.736i)8-s + (0.809 − 0.587i)9-s + (0.281 + 0.959i)12-s + (0.791 + 0.610i)13-s + (−0.998 − 0.0570i)16-s + (0.996 − 0.0855i)17-s + (−0.170 + 0.985i)18-s + (−0.921 − 0.389i)19-s + (−0.540 − 0.841i)23-s + (−0.870 − 0.491i)24-s + (−0.993 + 0.113i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)2-s + (−0.951 + 0.309i)3-s + (0.0285 − 0.999i)4-s + (0.466 − 0.884i)6-s + (0.676 + 0.736i)8-s + (0.809 − 0.587i)9-s + (0.281 + 0.959i)12-s + (0.791 + 0.610i)13-s + (−0.998 − 0.0570i)16-s + (0.996 − 0.0855i)17-s + (−0.170 + 0.985i)18-s + (−0.921 − 0.389i)19-s + (−0.540 − 0.841i)23-s + (−0.870 − 0.491i)24-s + (−0.993 + 0.113i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6855797889 + 0.05190774830i\)
\(L(\frac12)\) \(\approx\) \(0.6855797889 + 0.05190774830i\)
\(L(1)\) \(\approx\) \(0.5461638363 + 0.1612227836i\)
\(L(1)\) \(\approx\) \(0.5461638363 + 0.1612227836i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.717 + 0.696i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.791 + 0.610i)T \)
17 \( 1 + (0.996 - 0.0855i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (-0.941 + 0.336i)T \)
37 \( 1 + (-0.825 - 0.564i)T \)
41 \( 1 + (-0.897 + 0.441i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.170 + 0.985i)T \)
53 \( 1 + (0.0570 + 0.998i)T \)
59 \( 1 + (0.897 + 0.441i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (0.967 - 0.254i)T \)
79 \( 1 + (-0.198 - 0.980i)T \)
83 \( 1 + (0.931 - 0.362i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.491 + 0.870i)T \)
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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

−18.35439599780594519186932705745, −17.8267455862045859163986402273, −17.02131851547252501332599646921, −16.67222005002962901065356364671, −15.90735901430436657736635280052, −15.232495333888400314018863452401, −14.06030172890170743532635426420, −13.3067499624245567871568614122, −12.56839954335274593565995705215, −12.29343982088868160173883030750, −11.258214296493613531870915209432, −10.99201882401443895389404227119, −10.14532974099813219837620255625, −9.71095627896323475710088110400, −8.5825335979055998697117438729, −8.05395136639626394410597488441, −7.28108364198690010103003849966, −6.59140834726501716382983341439, −5.67502258600789963294323395338, −5.0792134632561423279962615995, −3.7950857085956253870303808167, −3.52863100604270926590518732361, −2.141462361640777524087383402507, −1.55764383724745580036854991661, −0.68264106419301659418427244299, 0.44363994204402585568272558495, 1.383397840494813224584432905219, 2.23219452325505770138041510143, 3.70052542912918370507858211001, 4.389347992969254290868086936563, 5.213351880570718235614862840531, 5.94112939325470822063348290835, 6.41156849745043493543621964721, 7.16768999202062208319294349166, 7.90509087772088410255766581190, 8.84107355738275062140687455698, 9.32975161748186847912904246170, 10.23660954916378753634325573474, 10.68756416639406820529645138283, 11.333537481823596236819075693254, 12.11050111490948044555539866770, 12.884550526813192834546011946339, 13.8449691948340393619482573268, 14.49745063013673192491733068196, 15.284188603602480882732501652257, 15.846927515652328120222108775347, 16.54469815847756394910043946450, 16.88886064853467925240974495911, 17.61224115648349930561247318812, 18.32522863992368827637976279487

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