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Properties

Label 1-2496-2496.1805-r0-0-0
Degree $1$
Conductor $2496$
Sign $-0.447 + 0.894i$
Analytic cond. $11.5913$
Root an. cond. $11.5913$
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  + (−0.382 − 0.923i)5-s + (0.258 − 0.965i)7-s + (−0.130 + 0.991i)11-s + (0.866 − 0.5i)17-s + (−0.991 + 0.130i)19-s + (−0.965 + 0.258i)23-s + (−0.707 + 0.707i)25-s + (0.608 + 0.793i)29-s − i·31-s + (−0.991 + 0.130i)35-s + (−0.991 − 0.130i)37-s + (0.258 + 0.965i)41-s + (−0.608 + 0.793i)43-s − 47-s + (−0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)5-s + (0.258 − 0.965i)7-s + (−0.130 + 0.991i)11-s + (0.866 − 0.5i)17-s + (−0.991 + 0.130i)19-s + (−0.965 + 0.258i)23-s + (−0.707 + 0.707i)25-s + (0.608 + 0.793i)29-s − i·31-s + (−0.991 + 0.130i)35-s + (−0.991 − 0.130i)37-s + (0.258 + 0.965i)41-s + (−0.608 + 0.793i)43-s − 47-s + (−0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (1805, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1282636603 + 0.2076255170i\)
\(L(\frac12)\) \(\approx\) \(0.1282636603 + 0.2076255170i\)
\(L(1)\) \(\approx\) \(0.8003370860 - 0.1397693556i\)
\(L(1)\) \(\approx\) \(0.8003370860 - 0.1397693556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.991 + 0.130i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (0.258 + 0.965i)T \)
43 \( 1 + (-0.608 + 0.793i)T \)
47 \( 1 - T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (0.608 - 0.793i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.793 + 0.608i)T \)
71 \( 1 + (0.258 - 0.965i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−19.35544145584723593147846940615, −18.533412415970826573576725808351, −18.00445564590046879025809613645, −17.16080402851190538981470263818, −16.233627494136808936997861092941, −15.603330391539993836691813484854, −14.96469185082302374437451971675, −14.28262295509274955428931991330, −13.69793892916210170721495058860, −12.562348417690170809368830801336, −11.97257878996573790694191645491, −11.33316523621543606880531150372, −10.48164452251938322229448888616, −10.02648614076216332994406906764, −8.637984354688906556930146799236, −8.43373644039185563688296394482, −7.51206681897197818995375956000, −6.499980635147822166237725960715, −5.979249965191988166774717358385, −5.169373521765765822251212215978, −4.03589743395079080264117758348, −3.29083690533739747708006852156, −2.531069121480294987652621623886, −1.66937647692378981451363282952, −0.078506490335927813576908248686, 1.182546968201495222492286287738, 1.86744202785723392628357198013, 3.16616132366592437948894933815, 4.157108389461696721613269049996, 4.58397331936292415319173872903, 5.39408488928474869699950782119, 6.444402918132529110489007700030, 7.37521092709780169231991708952, 7.89583294113273623309102678182, 8.606174963589823353450021324463, 9.677759314685442561679505182649, 10.091038727412039236029799106686, 11.05187068783235286216941689730, 11.87583271358504066664137135978, 12.50078064078231773153470739872, 13.15692716615775844970310393524, 13.93908639320523051099162828862, 14.71012260184340794640048035782, 15.41680629438411879136252843322, 16.39995336217868402671419267774, 16.66128969532279102715621489996, 17.562273749661017850602779407474, 18.07699194897748535312136564098, 19.17285626754754371964571367228, 19.809549751804688130145336577440

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