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Properties

Label 2-369-9.4-c1-0-29
Degree $2$
Conductor $369$
Sign $-0.0500 + 0.998i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
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  + (0.178 + 0.309i)2-s + (−1.39 − 1.02i)3-s + (0.935 − 1.62i)4-s + (1.28 − 2.21i)5-s + (0.0694 − 0.616i)6-s + (0.644 + 1.11i)7-s + 1.38·8-s + (0.883 + 2.86i)9-s + 0.917·10-s + (−1.06 − 1.84i)11-s + (−2.97 + 1.29i)12-s + (0.743 − 1.28i)13-s + (−0.230 + 0.399i)14-s + (−4.06 + 1.77i)15-s + (−1.62 − 2.81i)16-s − 2.07·17-s + ⋯
L(s)  = 1  + (0.126 + 0.219i)2-s + (−0.804 − 0.593i)3-s + (0.467 − 0.810i)4-s + (0.573 − 0.992i)5-s + (0.0283 − 0.251i)6-s + (0.243 + 0.421i)7-s + 0.489·8-s + (0.294 + 0.955i)9-s + 0.290·10-s + (−0.321 − 0.556i)11-s + (−0.857 + 0.374i)12-s + (0.206 − 0.357i)13-s + (−0.0616 + 0.106i)14-s + (−1.05 + 0.458i)15-s + (−0.405 − 0.703i)16-s − 0.503·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0500 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0500 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.0500 + 0.998i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ -0.0500 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921418 - 0.968769i\)
\(L(\frac12)\) \(\approx\) \(0.921418 - 0.968769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 + 1.02i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.178 - 0.309i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.28 + 2.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.644 - 1.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.06 + 1.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.743 + 1.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + 0.0249T + 19T^{2} \)
23 \( 1 + (-0.409 + 0.710i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.48 + 6.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.48 - 7.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
43 \( 1 + (-5.30 - 9.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.08 + 5.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.89T + 53T^{2} \)
59 \( 1 + (5.88 - 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 8.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 + 4.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.08T + 71T^{2} \)
73 \( 1 - 4.52T + 73T^{2} \)
79 \( 1 + (-1.66 - 2.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.93 - 3.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.51T + 89T^{2} \)
97 \( 1 + (-0.992 - 1.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.18262560371286640799670677915, −10.45927869939744185618019717979, −9.356403001369614379882353479122, −8.309833878424076460890193912396, −7.18587190858397793991412727066, −6.01556653914908904882054014347, −5.56162472382292143344580842230, −4.69067575867431833967203476424, −2.23796817874833157156705730687, −1.00718627998991920339924849208, 2.17788452747499360901538715457, 3.55556084515264691435875299971, 4.55162565232742646026959907571, 5.93916888246249569314746908498, 6.87173323011447403712770221216, 7.57889104789018199735253576829, 9.117217211239445079121478755118, 10.10402052991730652412284125027, 11.02210228082546425271634251994, 11.22246958720127903948245138941

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