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Properties

Label 2-418-209.151-c1-0-8
Degree $2$
Conductor $418$
Sign $0.734 + 0.678i$
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 + (−2.13 + 0.693i)3-s + (0.309 − 0.951i)4-s + (−0.363 − 0.264i)5-s + (−1.31 + 1.81i)6-s + (1.26 + 0.410i)7-s + (−0.309 − 0.951i)8-s + (1.64 − 1.19i)9-s − 0.449·10-s + (3.29 + 0.362i)11-s + 2.24i·12-s + (1.70 − 1.23i)13-s + (1.26 − 0.410i)14-s + (0.959 + 0.311i)15-s + (−0.809 − 0.587i)16-s + (2.44 − 3.36i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−1.23 + 0.400i)3-s + (0.154 − 0.475i)4-s + (−0.162 − 0.118i)5-s + (−0.538 + 0.741i)6-s + (0.477 + 0.155i)7-s + (−0.109 − 0.336i)8-s + (0.549 − 0.399i)9-s − 0.142·10-s + (0.994 + 0.109i)11-s + 0.647i·12-s + (0.471 − 0.342i)13-s + (0.337 − 0.109i)14-s + (0.247 + 0.0804i)15-s + (−0.202 − 0.146i)16-s + (0.593 − 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26449 - 0.494204i\)
\(L(\frac12)\) \(\approx\) \(1.26449 - 0.494204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.29 - 0.362i)T \)
19 \( 1 + (-0.334 + 4.34i)T \)
good3 \( 1 + (2.13 - 0.693i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.363 + 0.264i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.26 - 0.410i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.70 + 1.23i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.44 + 3.36i)T + (-5.25 - 16.1i)T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 + (0.875 - 2.69i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.26 - 8.62i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.74 - 1.21i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.23 + 6.88i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + (0.617 + 1.89i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.722 + 0.994i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (7.39 + 2.40i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.42 - 8.84i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.99iT - 67T^{2} \)
71 \( 1 + (-2.98 + 4.11i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.18 + 1.03i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.39 + 3.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.93 - 10.9i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.49iT - 89T^{2} \)
97 \( 1 + (-1.31 - 1.80i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.26510683340249740341683114738, −10.55680509250158483132619467890, −9.558519729878988726471826577432, −8.486890388229533887521202626077, −6.99587140224799410932453071209, −6.13161155371960979427425750132, −5.06423077714560385782072974301, −4.53277060621231454478487713891, −3.06691134148412189430008893440, −1.05586681172385712088672301331, 1.39981978259397288216897365561, 3.58390531431456648044559467708, 4.62715467782827546391677707155, 5.86344714524899324484275612493, 6.27876066201140787742976774807, 7.36360426091602862557159255310, 8.263292546616978851974699362777, 9.552934186113008614974874918945, 10.86404393614431898435469496579, 11.48497217668370256320478204998

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