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Properties

Label 2-416-52.31-c1-0-12
Degree $2$
Conductor $416$
Sign $0.105 + 0.994i$
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  − 1.46i·3-s + (1.87 − 1.87i)5-s + (−0.675 + 0.675i)7-s + 0.865·9-s + (0.202 − 0.202i)11-s + (0.675 − 3.54i)13-s + (−2.74 − 2.74i)15-s + 0.539i·17-s + (1.66 + 1.66i)19-s + (0.987 + 0.987i)21-s − 4.67·23-s − 2.05i·25-s − 5.64i·27-s − 4.27·29-s + (−5.66 − 5.66i)31-s + ⋯
L(s)  = 1  − 0.843i·3-s + (0.840 − 0.840i)5-s + (−0.255 + 0.255i)7-s + 0.288·9-s + (0.0610 − 0.0610i)11-s + (0.187 − 0.982i)13-s + (−0.708 − 0.708i)15-s + 0.130i·17-s + (0.381 + 0.381i)19-s + (0.215 + 0.215i)21-s − 0.975·23-s − 0.411i·25-s − 1.08i·27-s − 0.793·29-s + (−1.01 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16345 - 1.04683i\)
\(L(\frac12)\) \(\approx\) \(1.16345 - 1.04683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.675 + 3.54i)T \)
good3 \( 1 + 1.46iT - 3T^{2} \)
5 \( 1 + (-1.87 + 1.87i)T - 5iT^{2} \)
7 \( 1 + (0.675 - 0.675i)T - 7iT^{2} \)
11 \( 1 + (-0.202 + 0.202i)T - 11iT^{2} \)
17 \( 1 - 0.539iT - 17T^{2} \)
19 \( 1 + (-1.66 - 1.66i)T + 19iT^{2} \)
23 \( 1 + 4.67T + 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + (5.66 + 5.66i)T + 31iT^{2} \)
37 \( 1 + (-4.80 - 4.80i)T + 37iT^{2} \)
41 \( 1 + (-2.21 + 2.21i)T - 41iT^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (-1.08 + 1.08i)T - 47iT^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + (3.41 - 3.41i)T - 59iT^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-9.28 - 9.28i)T + 67iT^{2} \)
71 \( 1 + (-9.35 - 9.35i)T + 71iT^{2} \)
73 \( 1 + (0.865 + 0.865i)T + 73iT^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 + (3.12 + 3.12i)T + 83iT^{2} \)
89 \( 1 + (-6.21 - 6.21i)T + 89iT^{2} \)
97 \( 1 + (7.67 - 7.67i)T - 97iT^{2} \)
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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

−11.03224177334233226873087137298, −9.893675527494367502486952634909, −9.277122760457355233903054772938, −8.130256295032696526488744842991, −7.39003014870538152324338170923, −6.03464451819148946773445799309, −5.57368512298982810965494114922, −4.06042218282149418101858970566, −2.36147677493962928224413752350, −1.12830976700908972340298573837, 2.02763062295960104066197859786, 3.47843798605697108462341812610, 4.46303085378377621533296059061, 5.73492316597641801199723614566, 6.68801935898461916320702187490, 7.55190177506152196743057163873, 9.239643231876698838438507005825, 9.517325926831623922685703245907, 10.56900839178756947667521755153, 11.00998775233045967003268331150

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