↑

Properties

Label 1-2496-2496.419-r0-0-0
Degree $1$
Conductor $2496$
Sign $0.577 + 0.816i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 + 0.923i)5-s + (0.965 + 0.258i)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 + (−0.793 + 0.608i)29-s + 31-s + (0.130 + 0.991i)35-s + (−0.991 − 0.130i)37-s + (−0.965 + 0.258i)41-s + (0.793 + 0.608i)43-s − i·47-s + (0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)5-s + (0.965 + 0.258i)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 + (−0.793 + 0.608i)29-s + 31-s + (0.130 + 0.991i)35-s + (−0.991 − 0.130i)37-s + (−0.965 + 0.258i)41-s + (0.793 + 0.608i)43-s − i·47-s + (0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.853272971 + 0.9594633279i\)
\(L(\frac12)\) \(\approx\) \(1.853272971 + 0.9594633279i\)
\(L(1)\) \(\approx\) \(1.281824517 + 0.2698662039i\)
\(L(1)\) \(\approx\) \(1.281824517 + 0.2698662039i\)

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 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (-0.793 + 0.608i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (-0.965 + 0.258i)T \)
43 \( 1 + (0.793 + 0.608i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (-0.608 + 0.793i)T \)
61 \( 1 + (-0.130 - 0.991i)T \)
67 \( 1 + (0.793 - 0.608i)T \)
71 \( 1 + (-0.965 - 0.258i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (0.258 + 0.965i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.50012074494179304132353558932, −18.47249975981667700737164688683, −17.77390994231561242856312334524, −17.2516762775023805506537655716, −16.74526572021757962616278144388, −15.61573660495568922770243784013, −15.25712539456640085001393611139, −14.2010948034199157467451129813, −13.6043166344771692160701620628, −13.00653278026202868904562284416, −11.94106617879843908610443617784, −11.68678061018749612155773801030, −10.57929759271980945719840601405, −9.85218358945050691288117263315, −9.06524743841033731862725891727, −8.506685611811674244236706578686, −7.5006159941902277718849295902, −7.010362431656901636138651986390, −5.80175784402900665285533962255, −4.98106584550450611313039515391, −4.62908990780858015110281360284, −3.652335595462833267793339307428, −2.31986182895972981901775436825, −1.6785201692050463563314244600, −0.75857050656546525185133927219, 1.0791914314293108872570799991, 2.00195135671536506199203598394, 2.886293910163172161614803881375, 3.59030284606778642314061729189, 4.71648919809595752780026378615, 5.47311593521848112832436408597, 6.24640595291377407243143850043, 6.99964374188936651257957202853, 7.80831608534526046437603151023, 8.64504884032547626054202884095, 9.27044944693462837092399790564, 10.29976783922889834170777408896, 11.08412585898549559462463305909, 11.29624770326118169515363531391, 12.28417655879254573292834450896, 13.35194518282599526904984186002, 13.8876901329349154041455561519, 14.539771176571386360272249726688, 15.2093056826400558856889994411, 15.86262200333126988818530725205, 16.92489992488297056537225612602, 17.4967672228766278683900067915, 18.18848806618121130000662179029, 18.75862042178155454385239135199, 19.41190988297176569434755432895

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