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Properties

Label 2-418-209.208-c1-0-7
Degree $2$
Conductor $418$
Sign $0.938 + 0.345i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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-s + 1.58i·3-s + 4-s − 3.32·5-s − 1.58i·6-s − 2.41i·7-s − 8-s + 0.483·9-s + 3.32·10-s + (−1.04 − 3.14i)11-s + 1.58i·12-s + 3.85·13-s + 2.41i·14-s − 5.26i·15-s + 16-s + 1.39i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.915i·3-s + 0.5·4-s − 1.48·5-s − 0.647i·6-s − 0.914i·7-s − 0.353·8-s + 0.161·9-s + 1.05·10-s + (−0.314 − 0.949i)11-s + 0.457i·12-s + 1.07·13-s + 0.646i·14-s − 1.36i·15-s + 0.250·16-s + 0.337i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746344 - 0.132874i\)
\(L(\frac12)\) \(\approx\) \(0.746344 - 0.132874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 + (1.04 + 3.14i)T \)
19 \( 1 + (-2.71 + 3.41i)T \)
good3 \( 1 - 1.58iT - 3T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 + 2.41iT - 7T^{2} \)
13 \( 1 - 3.85T + 13T^{2} \)
17 \( 1 - 1.39iT - 17T^{2} \)
23 \( 1 - 5.55T + 23T^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 + 5.70iT - 37T^{2} \)
41 \( 1 + 2.95T + 41T^{2} \)
43 \( 1 + 4.05iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 7.99iT - 71T^{2} \)
73 \( 1 + 5.49iT - 73T^{2} \)
79 \( 1 + 3.59T + 79T^{2} \)
83 \( 1 + 1.78iT - 83T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 - 4.62iT - 97T^{2} \)
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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

−10.94057550800698378528007654312, −10.51896372201704440976105469133, −9.246414661696288789909533790880, −8.497453735049580920300511640014, −7.58736444299034263166999890247, −6.81190860795510204624944034991, −5.19779152037793208245951230545, −3.91810145506553482724426769677, −3.40421306270928754850830862151, −0.74175404979455627038696214270, 1.27603212719977215711898238829, 2.83482269006274851332282380473, 4.25810879321570677501817729954, 5.77433624778691430361377935066, 7.03321508119472222057446871606, 7.55925315636334731786236286225, 8.347292275452378700603622715802, 9.203224158917698353918254235847, 10.41949601475226860048948962833, 11.45589007508691310852216407488

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