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Properties

Label 2-416-52.43-c2-0-5
Degree $2$
Conductor $416$
Sign $0.868 - 0.496i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.33 − 1.34i)3-s + 8.92i·5-s + (−5.61 − 9.72i)7-s + (−0.863 − 1.49i)9-s + (2.08 − 3.60i)11-s + (3.58 − 12.4i)13-s + (12.0 − 20.8i)15-s + (12.4 + 21.5i)17-s + (15.7 + 27.2i)19-s + 30.2i·21-s + (18.4 + 10.6i)23-s − 54.5·25-s + 28.9i·27-s + (5.64 − 9.77i)29-s − 16.1·31-s + ⋯
L(s)  = 1  + (−0.778 − 0.449i)3-s + 1.78i·5-s + (−0.802 − 1.38i)7-s + (−0.0959 − 0.166i)9-s + (0.189 − 0.327i)11-s + (0.275 − 0.961i)13-s + (0.801 − 1.38i)15-s + (0.731 + 1.26i)17-s + (0.828 + 1.43i)19-s + 1.44i·21-s + (0.801 + 0.462i)23-s − 2.18·25-s + 1.07i·27-s + (0.194 − 0.336i)29-s − 0.522·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ 0.868 - 0.496i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05355 + 0.279817i\)
\(L(\frac12)\) \(\approx\) \(1.05355 + 0.279817i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-3.58 + 12.4i)T \)
good3 \( 1 + (2.33 + 1.34i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 - 8.92iT - 25T^{2} \)
7 \( 1 + (5.61 + 9.72i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.08 + 3.60i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (-12.4 - 21.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-15.7 - 27.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.4 - 10.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.64 + 9.77i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 16.1T + 961T^{2} \)
37 \( 1 + (-32.9 - 19.0i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-61.8 - 35.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-14.5 + 8.37i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 16.2T + 2.20e3T^{2} \)
53 \( 1 - 45.8T + 2.80e3T^{2} \)
59 \( 1 + (-26.8 - 46.5i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.39 - 4.14i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-27.2 + 47.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (51.1 + 88.6i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 42.5iT - 5.32e3T^{2} \)
79 \( 1 + 68.1iT - 6.24e3T^{2} \)
83 \( 1 - 26.4T + 6.88e3T^{2} \)
89 \( 1 + (-6.84 - 3.95i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-55.2 + 31.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86785938449723008149955163701, −10.45482781037432117096974730113, −9.691582779121976158661101701044, −7.81374642754663043960217331938, −7.28280890628721581147016652805, −6.21125204836938399776467649362, −5.92492937145513058561216467727, −3.67879050804753138586270169636, −3.22873638577816718961161901099, −0.999666096948465058656188505077, 0.69395096703049861261673287813, 2.55384205429007851318016968709, 4.38824719003369763979994681072, 5.22459949249367635348406960095, 5.71047676299230556386588265236, 7.08066104584248519090617503140, 8.547922820008705147475369323353, 9.307724216668302580437393218067, 9.563997296474938441254193374197, 11.25841322753947085278190020084

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