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Properties

Label 2-416-13.9-c1-0-2
Degree $2$
Conductor $416$
Sign $0.202 - 0.979i$
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  − 2.46·5-s + (1.5 + 2.59i)9-s + (0.232 + 3.59i)13-s + (3.96 + 6.86i)17-s + 1.07·25-s + (−4.23 + 7.33i)29-s + (4.69 − 8.13i)37-s + (−0.964 + 1.66i)41-s + (−3.69 − 6.40i)45-s + (3.5 − 6.06i)49-s − 10.4·53-s + (2.69 + 4.66i)61-s + (−0.571 − 8.86i)65-s + 16.8·73-s + (−4.5 + 7.79i)81-s + ⋯
L(s)  = 1  − 1.10·5-s + (0.5 + 0.866i)9-s + (0.0643 + 0.997i)13-s + (0.961 + 1.66i)17-s + 0.214·25-s + (−0.785 + 1.36i)29-s + (0.772 − 1.33i)37-s + (−0.150 + 0.260i)41-s + (−0.550 − 0.954i)45-s + (0.5 − 0.866i)49-s − 1.43·53-s + (0.345 + 0.597i)61-s + (−0.0709 − 1.09i)65-s + 1.97·73-s + (−0.5 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795764 + 0.648116i\)
\(L(\frac12)\) \(\approx\) \(0.795764 + 0.648116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.232 - 3.59i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.96 - 6.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.23 - 7.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-4.69 + 8.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.964 - 1.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.69 - 4.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9 + 15.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.28351126672163938106742399776, −10.70706909143103643754701434867, −9.635914733686147124706346125353, −8.470873242182541545297031238478, −7.76117247492538438037530362028, −6.93822264004032859652818746512, −5.62213449270403166404704212725, −4.37047038976356805569766688107, −3.59302494709322558959505063756, −1.76216394319035606246726551869, 0.70193976957818435534057543417, 2.99872450239467610938363441290, 3.94482325744288031830849481082, 5.11063934362423115047210656723, 6.36346122064774768861193478682, 7.53790765312086117280878972105, 7.975445093240735173481378788063, 9.337112767649430387900379335932, 9.987957045803917386790591564137, 11.23358154645737432795751354783

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