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Properties

Label 2-418-11.4-c1-0-6
Degree $2$
Conductor $418$
Sign $0.339 - 0.940i$
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  + (0.809 + 0.587i)2-s + (0.388 − 1.19i)3-s + (0.309 + 0.951i)4-s + (−1.36 + 0.989i)5-s + (1.01 − 0.738i)6-s + (1.33 + 4.11i)7-s + (−0.309 + 0.951i)8-s + (1.14 + 0.834i)9-s − 1.68·10-s + (−1.19 + 3.09i)11-s + 1.25·12-s + (−4.20 − 3.05i)13-s + (−1.33 + 4.11i)14-s + (0.653 + 2.01i)15-s + (−0.809 + 0.587i)16-s + (5.19 − 3.77i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.224 − 0.690i)3-s + (0.154 + 0.475i)4-s + (−0.608 + 0.442i)5-s + (0.415 − 0.301i)6-s + (0.505 + 1.55i)7-s + (−0.109 + 0.336i)8-s + (0.382 + 0.278i)9-s − 0.532·10-s + (−0.361 + 0.932i)11-s + 0.362·12-s + (−1.16 − 0.847i)13-s + (−0.357 + 1.09i)14-s + (0.168 + 0.519i)15-s + (−0.202 + 0.146i)16-s + (1.26 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52818 + 1.07311i\)
\(L(\frac12)\) \(\approx\) \(1.52818 + 1.07311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.19 - 3.09i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.388 + 1.19i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.36 - 0.989i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.33 - 4.11i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.20 + 3.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.19 + 3.77i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + (-2.19 - 6.76i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.80 - 2.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.48 + 4.56i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.10 + 6.48i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + (-3.72 + 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.09 + 4.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.21 + 12.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.31 - 6.04i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.78T + 67T^{2} \)
71 \( 1 + (-3.04 + 2.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.51 - 4.65i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.21 + 0.879i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.68 - 4.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.194T + 89T^{2} \)
97 \( 1 + (2.80 + 2.03i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.87118935900375393781953113163, −10.59191629768025161575802641167, −9.455588530484646167577585860151, −8.241901701391234982123200189916, −7.49892835141486714188827082498, −7.00839543746102815763093681239, −5.39373442553921352658630424588, −4.94089323110985403824876425491, −3.10257744264926081730775369335, −2.18957557511263114733952925374, 1.08662420325137212404678302115, 3.19010764907287487234156182397, 4.33480454784479948165859258331, 4.52327083774268126802372441417, 6.13195692571758538260297561312, 7.46690559330963104628805966608, 8.151278597497373671578784491472, 9.575433320422505381314317295769, 10.21421319456620402667249750192, 10.97760343628973465816004568340

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