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Properties

Label 2-416-104.101-c1-0-4
Degree $2$
Conductor $416$
Sign $0.837 + 0.545i$
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  + (−2.28 + 1.31i)3-s − 3.40·5-s + (−1.30 − 0.754i)7-s + (1.97 − 3.42i)9-s + (1.69 + 2.93i)11-s + (2.16 − 2.88i)13-s + (7.77 − 4.49i)15-s + (−1.32 + 2.29i)17-s + (3.32 − 5.76i)19-s + 3.97·21-s + (0.307 + 0.532i)23-s + 6.60·25-s + 2.50i·27-s + (2.50 − 1.44i)29-s + 0.813i·31-s + ⋯
L(s)  = 1  + (−1.31 + 0.761i)3-s − 1.52·5-s + (−0.494 − 0.285i)7-s + (0.658 − 1.14i)9-s + (0.511 + 0.885i)11-s + (0.599 − 0.800i)13-s + (2.00 − 1.15i)15-s + (−0.322 + 0.557i)17-s + (0.763 − 1.32i)19-s + 0.868·21-s + (0.0640 + 0.110i)23-s + 1.32·25-s + 0.482i·27-s + (0.465 − 0.268i)29-s + 0.146i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.837 + 0.545i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.837 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461688 - 0.137090i\)
\(L(\frac12)\) \(\approx\) \(0.461688 - 0.137090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 2.88i)T \)
good3 \( 1 + (2.28 - 1.31i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
7 \( 1 + (1.30 + 0.754i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.69 - 2.93i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.32 - 2.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.307 - 0.532i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.813iT - 31T^{2} \)
37 \( 1 + (-2.53 - 4.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.98 + 4.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.93 + 4.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.88iT - 47T^{2} \)
53 \( 1 + 0.627iT - 53T^{2} \)
59 \( 1 + (-1.23 + 2.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.75 + 3.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.664 - 1.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.38 - 3.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 1.65T + 79T^{2} \)
83 \( 1 - 7.81T + 83T^{2} \)
89 \( 1 + (1.12 - 0.648i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.5 + 7.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.18447439812348122721757972740, −10.46664335815420939323230518576, −9.559503798395606687387405054632, −8.354388788071000313067474869304, −7.22223237404429123618070319212, −6.43525488119887272907789765854, −5.12838056278003306550971828760, −4.30275540057952240821374265209, −3.43094534666818346575402769255, −0.48643170129804691581611459366, 1.03738045061200213345938579083, 3.34760315830660484646409381039, 4.46100059570363427494921411849, 5.82911613500780456518035162194, 6.51911105773819746700974464385, 7.44520701611503914944686446922, 8.327659763630343109339557357688, 9.464852187354665522775176287498, 10.98805168111229664561068082313, 11.37243439244874561498117181616

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