Properties

Label 2-416-13.9-c1-0-7
Degree $2$
Conductor $416$
Sign $0.981 - 0.189i$
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  + 4.46·5-s + (1.5 + 2.59i)9-s + (−3.23 + 1.59i)13-s + (−2.96 − 5.13i)17-s + 14.9·25-s + (−0.767 + 1.33i)29-s + (−5.69 + 9.86i)37-s + (5.96 − 10.3i)41-s + (6.69 + 11.5i)45-s + (3.5 − 6.06i)49-s − 3.53·53-s + (−7.69 − 13.3i)61-s + (−14.4 + 7.13i)65-s − 10.8·73-s + (−4.5 + 7.79i)81-s + ⋯
L(s)  = 1  + 1.99·5-s + (0.5 + 0.866i)9-s + (−0.896 + 0.443i)13-s + (−0.718 − 1.24i)17-s + 2.98·25-s + (−0.142 + 0.246i)29-s + (−0.936 + 1.62i)37-s + (0.931 − 1.61i)41-s + (0.998 + 1.72i)45-s + (0.5 − 0.866i)49-s − 0.485·53-s + (−0.985 − 1.70i)61-s + (−1.78 + 0.884i)65-s − 1.27·73-s + (−0.5 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.981 - 0.189i$
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: \((2,\ 416,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84949 + 0.177150i\)
\(L(\frac12)\) \(\approx\) \(1.84949 + 0.177150i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 - 1.59i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 4.46T + 5T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.96 + 5.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.767 - 1.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (5.69 - 9.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.96 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 3.53T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.69 + 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
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

−11.00349800252223220338889326889, −10.17906569591599664136860016188, −9.560763677699288361840039651690, −8.790906528279219755841431238019, −7.29225986188843574284946001380, −6.56546368161579126123375522011, −5.34646869418283280224301128829, −4.75217347570088163709179678261, −2.64965181453340557889179221113, −1.79290543314879756681242465962, 1.55645804680609255556160859075, 2.73588033518958669744182930141, 4.39491055892099830466380941802, 5.67140035079707015491724778430, 6.26975979927335576912515009282, 7.26904459171443857326305880975, 8.769313739828517293082285312918, 9.479259206124106537865589622193, 10.15226851265646105006239828246, 10.88557293000375298828973814736

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