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Properties

Label 1-4235-4235.964-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.926 + 0.376i$
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  + (−0.398 − 0.917i)2-s + (0.913 − 0.406i)3-s + (−0.683 + 0.730i)4-s + (−0.736 − 0.676i)6-s + (0.941 + 0.336i)8-s + (0.669 − 0.743i)9-s + (−0.327 + 0.945i)12-s + (−0.516 + 0.856i)13-s + (−0.0665 − 0.997i)16-s + (−0.161 − 0.986i)17-s + (−0.948 − 0.318i)18-s + (0.449 + 0.893i)19-s + (−0.928 + 0.371i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.398 − 0.917i)2-s + (0.913 − 0.406i)3-s + (−0.683 + 0.730i)4-s + (−0.736 − 0.676i)6-s + (0.941 + 0.336i)8-s + (0.669 − 0.743i)9-s + (−0.327 + 0.945i)12-s + (−0.516 + 0.856i)13-s + (−0.0665 − 0.997i)16-s + (−0.161 − 0.986i)17-s + (−0.948 − 0.318i)18-s + (0.449 + 0.893i)19-s + (−0.928 + 0.371i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.926 + 0.376i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (964, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.926 + 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1486850971 - 0.7617696013i\)
\(L(\frac12)\) \(\approx\) \(-0.1486850971 - 0.7617696013i\)
\(L(1)\) \(\approx\) \(0.7820643322 - 0.5172683508i\)
\(L(1)\) \(\approx\) \(0.7820643322 - 0.5172683508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.398 - 0.917i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.516 + 0.856i)T \)
17 \( 1 + (-0.161 - 0.986i)T \)
19 \( 1 + (0.449 + 0.893i)T \)
23 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.362 - 0.931i)T \)
31 \( 1 + (-0.797 + 0.603i)T \)
37 \( 1 + (0.905 - 0.424i)T \)
41 \( 1 + (-0.870 + 0.491i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.948 + 0.318i)T \)
53 \( 1 + (0.0665 - 0.997i)T \)
59 \( 1 + (-0.861 + 0.508i)T \)
61 \( 1 + (-0.595 - 0.803i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (-0.999 - 0.0380i)T \)
79 \( 1 + (0.851 - 0.524i)T \)
83 \( 1 + (0.985 + 0.170i)T \)
89 \( 1 + (-0.0475 + 0.998i)T \)
97 \( 1 + (-0.564 - 0.825i)T \)
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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

−18.59892506403410651204536934020, −18.11304036551419441938731961411, −17.30435103278151873638083229266, −16.638034545931213420286262306235, −15.952249126146672011216734513, −15.31944320688577546550350672541, −14.813737346123415375254307988580, −14.323266812384648288140671456153, −13.3345705255317930887971336772, −13.106951491563968828214853780037, −12.02494585031042829196049556295, −10.78768653842825770695276308581, −10.32795248296404402417357974550, −9.66685187494166268592026750376, −8.955614092645477116486831695357, −8.33899531721758491124363571543, −7.75647689470908316249582584198, −7.10374417888471144745672576743, −6.25742076265957050193402411600, −5.35830848474390405067846294883, −4.72479235491870657414557629523, −3.94570217579684971212744817783, −3.096596761422164071149982031052, −2.13269346094729996762368758891, −1.21638012727792751240918585099, 0.20829681180400314892688065395, 1.54080315649993685559243013124, 1.91109584775746947955460376040, 2.86923184414163550394485879997, 3.44861413034484750260710376396, 4.2839578185938196637217264501, 4.9644904434857596341499813920, 6.230684501859602619889184519186, 7.112913132657916200935374819376, 7.80091107515895810904663576333, 8.27698947286261235064288810045, 9.280501103232767526835997718255, 9.574113747297580059176676516209, 10.21414024673478197120236240982, 11.28752536643726864101145057050, 11.957089100635373185061832442288, 12.33866240704002420150324040361, 13.33006061232575244848083578641, 13.76889716011228313644680612434, 14.358075427814630024840534674312, 15.09626754551561609756295589000, 16.24658358737563625849149797647, 16.55450612052213309829000440648, 17.73645665616051024630440819198, 18.17090632248288589986412435359

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