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Properties

Label 1-2496-2496.779-r0-0-0
Degree $1$
Conductor $2496$
Sign $-0.773 + 0.634i$
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 + (0.707 + 0.707i)7-s + (0.923 + 0.382i)11-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.923 − 0.382i)29-s − 31-s + (−0.923 + 0.382i)35-s + (−0.382 + 0.923i)37-s + (−0.707 + 0.707i)41-s + (−0.923 − 0.382i)43-s + i·47-s + i·49-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)5-s + (0.707 + 0.707i)7-s + (0.923 + 0.382i)11-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.923 − 0.382i)29-s − 31-s + (−0.923 + 0.382i)35-s + (−0.382 + 0.923i)37-s + (−0.707 + 0.707i)41-s + (−0.923 − 0.382i)43-s + i·47-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ -0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5031525324 + 1.406217124i\)
\(L(\frac12)\) \(\approx\) \(0.5031525324 + 1.406217124i\)
\(L(1)\) \(\approx\) \(0.9746843841 + 0.4781448035i\)
\(L(1)\) \(\approx\) \(0.9746843841 + 0.4781448035i\)

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 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (0.382 - 0.923i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (0.923 - 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + T \)
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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

−19.49306644545426697062724548059, −18.29149569166305599290742050743, −17.77569675149415930449098663291, −16.9062214831515089281150824237, −16.43061295924764043728205458798, −15.79847515945291786749737927996, −14.82104879525342107235152424214, −14.07624891860348507144203640428, −13.5641157107731821108556624408, −12.69220342575207985721963137140, −11.72855443691157464865289352408, −11.53238982130389753528372194914, −10.50443497925191978063121851770, −9.654981766737640145717046623727, −8.71672038236456553080561411683, −8.438476540469653930331885115105, −7.26036804951869098902140671429, −6.89685047239574342079317865946, −5.56536661561733690183359717653, −4.94667152840612794048530558566, −4.172666937425964966723684392059, −3.53730976524448303825443938102, −2.25749967290691952837057795764, −1.22830336756125391437671129492, −0.50808723685549554006920187746, 1.48331887108025226375885759981, 2.03996527001253839723599232487, 3.21793289531608061714821225806, 3.84797201047835523175551742523, 4.73611250206976932738331189035, 5.79471444650749899776771865216, 6.37128694845281523397051983135, 7.25359746684843416586404417155, 8.05980104582847423232274372781, 8.59801929111229376145200475650, 9.710101535737817593216197656129, 10.26561869623952243988148556738, 11.20698406350137104546646202042, 11.84055287081449759225962929476, 12.22990549440402294949339267792, 13.36513558596605219833104587575, 14.40012006384925560611820871427, 14.580321366360610089235066640946, 15.37124190714171472850409595483, 16.002454648121294959231868344986, 17.09945571215646252928434919212, 17.619102257782264987669957087710, 18.42540320588406872816067579565, 18.893541279370750318535271463358, 19.71848974737110091397442118683

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