Properties

Degree 2
Conductor 13
Sign $0.957 - 0.289i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.957 - 0.289i$
motivic weight  =  \(3\)
character  :  $\chi_{13} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 13,\ (\ :3/2),\ 0.957 - 0.289i)\)
\(L(2)\)  \(\approx\)  \(1.04076 + 0.154029i\)
\(L(\frac12)\)  \(\approx\)  \(1.04076 + 0.154029i\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 2. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-45.5 - 11.2i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 1.73iT - 125T^{2} \)
7 \( 1 + (12 - 6.92i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-12 - 6.92i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (58.5 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (99 - 57.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-39 + 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.5 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + (124.5 + 71.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-235.5 - 135. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (52 + 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 301. iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (246 - 142. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (681 + 393. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-915 + 528. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 458. iT - 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 789. iT - 5.71e5T^{2} \)
89 \( 1 + (846 + 488. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-174 + 100. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

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 $Z$-function along the critical line