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Properties

Label 2-13-13.4-c3-0-0
Degree 22
Conductor 1313
Sign 0.957−0.289i0.957 - 0.289i
Analytic cond. 0.7670240.767024
Root an. cond. 0.8757990.875799
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−2.5 − 4.33i)4-s − 1.73i·5-s + (−3 + 1.73i)6-s + (−12 + 6.92i)7-s − 22.5i·8-s + (11.5 + 19.9i)9-s + (1.49 − 2.59i)10-s + (12 + 6.92i)11-s + 10·12-s + (45.5 + 11.2i)13-s − 24·14-s + (2.99 + 1.73i)15-s + (−0.500 + 0.866i)16-s + (−58.5 − 101. i)17-s + ⋯
L(s)  = 1  + (0.530 + 0.306i)2-s + (−0.192 + 0.333i)3-s + (−0.312 − 0.541i)4-s − 0.154i·5-s + (−0.204 + 0.117i)6-s + (−0.647 + 0.374i)7-s − 0.995i·8-s + (0.425 + 0.737i)9-s + (0.0474 − 0.0821i)10-s + (0.328 + 0.189i)11-s + 0.240·12-s + (0.970 + 0.240i)13-s − 0.458·14-s + (0.0516 + 0.0298i)15-s + (−0.00781 + 0.0135i)16-s + (−0.834 − 1.44i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.957−0.289i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+3/2) L(s)=((0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.957−0.289i0.957 - 0.289i
Analytic conductor: 0.7670240.767024
Root analytic conductor: 0.8757990.875799
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ13(4,⋅)\chi_{13} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :3/2), 0.957−0.289i)(2,\ 13,\ (\ :3/2),\ 0.957 - 0.289i)

Particular Values

L(2)L(2) ≈\approx 1.04076+0.154029i1.04076 + 0.154029i
L(12)L(\frac12) ≈\approx 1.04076+0.154029i1.04076 + 0.154029i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1+(−45.5−11.2i)T 1 + (-45.5 - 11.2i)T
good2 1+(−1.5−0.866i)T+(4+6.92i)T2 1 + (-1.5 - 0.866i)T + (4 + 6.92i)T^{2}
3 1+(1−1.73i)T+(−13.5−23.3i)T2 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2}
5 1+1.73iT−125T2 1 + 1.73iT - 125T^{2}
7 1+(12−6.92i)T+(171.5−297.i)T2 1 + (12 - 6.92i)T + (171.5 - 297. i)T^{2}
11 1+(−12−6.92i)T+(665.5+1.15e3i)T2 1 + (-12 - 6.92i)T + (665.5 + 1.15e3i)T^{2}
17 1+(58.5+101.i)T+(−2.45e3+4.25e3i)T2 1 + (58.5 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(99−57.1i)T+(3.42e3−5.94e3i)T2 1 + (99 - 57.1i)T + (3.42e3 - 5.94e3i)T^{2}
23 1+(−39+67.5i)T+(−6.08e3−1.05e4i)T2 1 + (-39 + 67.5i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(−70.5+122.i)T+(−1.21e4−2.11e4i)T2 1 + (-70.5 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1−155.iT−2.97e4T2 1 - 155. iT - 2.97e4T^{2}
37 1+(124.5+71.8i)T+(2.53e4+4.38e4i)T2 1 + (124.5 + 71.8i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+(−235.5−135.i)T+(3.44e4+5.96e4i)T2 1 + (-235.5 - 135. i)T + (3.44e4 + 5.96e4i)T^{2}
43 1+(52+90.0i)T+(−3.97e4+6.88e4i)T2 1 + (52 + 90.0i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1−301.iT−1.03e5T2 1 - 301. iT - 1.03e5T^{2}
53 1−93T+1.48e5T2 1 - 93T + 1.48e5T^{2}
59 1+(246−142.i)T+(1.02e5−1.77e5i)T2 1 + (246 - 142. i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(72.5+125.i)T+(−1.13e5+1.96e5i)T2 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(681+393.i)T+(1.50e5+2.60e5i)T2 1 + (681 + 393. i)T + (1.50e5 + 2.60e5i)T^{2}
71 1+(−915+528.i)T+(1.78e5−3.09e5i)T2 1 + (-915 + 528. i)T + (1.78e5 - 3.09e5i)T^{2}
73 1−458.iT−3.89e5T2 1 - 458. iT - 3.89e5T^{2}
79 1−1.27e3T+4.93e5T2 1 - 1.27e3T + 4.93e5T^{2}
83 1+789.iT−5.71e5T2 1 + 789. iT - 5.71e5T^{2}
89 1+(846+488.i)T+(3.52e5+6.10e5i)T2 1 + (846 + 488. i)T + (3.52e5 + 6.10e5i)T^{2}
97 1+(−174+100.i)T+(4.56e5−7.90e5i)T2 1 + (-174 + 100. i)T + (4.56e5 - 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.38787102262126019114903646140, −18.38745997698692694960953777081, −16.39943340649917061283650070128, −15.52482972107435867423016421077, −13.97966085743690877963176414599, −12.76776063570069867432355301239, −10.67577508221439449057168113284, −9.145887368596140060017474345389, −6.44642247865807450236577898176, −4.61610486448213139359214098709, 3.82089278567738814603334571403, 6.55954385010205411763053889022, 8.751724509032825632034212394218, 10.94872441274391514249844106255, 12.62060054542997103347677841983, 13.36238127724265163184551261728, 15.14413292162704069430548515255, 16.92096272912926110469252774589, 18.00079251783951060487176370950, 19.50752909096018256887140678710

Graph of the ZZ-function along the critical line