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Properties

Label 2-416-13.3-c1-0-12
Degree $2$
Conductor $416$
Sign $-0.477 + 0.878i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)3-s + 5-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s + (2.5 − 2.59i)13-s + (−1 − 1.73i)15-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)19-s + (−1 − 1.73i)23-s − 4·25-s − 4.00·27-s + (−2.5 − 4.33i)29-s + 2·31-s + (−3.99 + 6.92i)33-s + (−2.5 − 4.33i)37-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + 0.447·5-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s + (0.693 − 0.720i)13-s + (−0.258 − 0.447i)15-s + (−0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s + (−0.208 − 0.361i)23-s − 0.800·25-s − 0.769·27-s + (−0.464 − 0.804i)29-s + 0.359·31-s + (−0.696 + 1.20i)33-s + (−0.410 − 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.477 + 0.878i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542525 - 0.912455i\)
\(L(\frac12)\) \(\approx\) \(0.542525 - 0.912455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 13T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98322650556110006452846065851, −10.24188008281258316411059250208, −8.943373161667052862379020486968, −8.057165138715353796603730874785, −7.11200806205607986272802979645, −5.93636381038984198334203286701, −5.67612001872800744029860585719, −3.85092699556815660406124571196, −2.31192757469160141331754803335, −0.74079073947940448674303916649, 2.01982745446997270784132451189, 3.76521666680915787686574047205, 4.78933096749039358130717592127, 5.55272950821372477571782559729, 6.71005873146354384170307873527, 7.80572251307510414354091038004, 9.133719478554666529755097779734, 9.818620437023757232388838948827, 10.49968898232328623053280887750, 11.36303698360904672155054736461

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