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Properties

Label 2-363-33.17-c1-0-19
Degree $2$
Conductor $363$
Sign $-0.700 + 0.713i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (−0.363 − 1.11i)2-s + (−0.451 + 1.67i)3-s + (0.499 − 0.363i)4-s + (−0.363 − 0.118i)5-s + (2.03 − 0.103i)6-s + (−1.80 − 2.48i)7-s + (−2.48 − 1.80i)8-s + (−2.59 − 1.50i)9-s + 0.449i·10-s + (0.381 + 0.999i)12-s + (−0.690 + 0.224i)13-s + (−2.12 + 2.92i)14-s + (0.361 − 0.554i)15-s + (−0.736 + 2.26i)16-s + (1.31 − 4.04i)17-s + (−0.744 + 3.44i)18-s + ⋯
L(s)  = 1  + (−0.256 − 0.790i)2-s + (−0.260 + 0.965i)3-s + (0.249 − 0.181i)4-s + (−0.162 − 0.0527i)5-s + (0.830 − 0.0421i)6-s + (−0.683 − 0.941i)7-s + (−0.880 − 0.639i)8-s + (−0.864 − 0.502i)9-s + 0.141i·10-s + (0.110 + 0.288i)12-s + (−0.191 + 0.0622i)13-s + (−0.568 + 0.782i)14-s + (0.0932 − 0.143i)15-s + (−0.184 + 0.566i)16-s + (0.318 − 0.981i)17-s + (−0.175 + 0.812i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290920 - 0.693580i\)
\(L(\frac12)\) \(\approx\) \(0.290920 - 0.693580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.451 - 1.67i)T \)
11 \( 1 \)
good2 \( 1 + (0.363 + 1.11i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.363 + 0.118i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.80 + 2.48i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.690 - 0.224i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.31 + 4.04i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.54 + 2.12i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.23iT - 23T^{2} \)
29 \( 1 + (1.90 - 1.38i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.80 - 5.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.42 - 1.76i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.587 - 0.427i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + (-4.89 + 6.73i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.76 + 0.572i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.224 - 0.309i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.5 + 0.812i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 + (-5.06 - 1.64i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.52 + 7.60i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.78 - 1.22i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.95 - 6.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.527iT - 89T^{2} \)
97 \( 1 + (1.35 + 4.16i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.93023260798206854158935527432, −10.16876898386647407082596663533, −9.707810919405796892059857000027, −8.707294536411456696468074549311, −7.15101052779633317115881631079, −6.26459557939341492077994496937, −4.93016619105133314839667333584, −3.74692529353032790963007090986, −2.76896069087677943810417635973, −0.54258761289188424168772325784, 2.10726026587560049406927318552, 3.39635023676419577632719303102, 5.76734425060937175617462602609, 5.86931553095893089216119501787, 7.16454135731450034249697906395, 7.79152222262669075968105033711, 8.683697132657255262085732055957, 9.676150963774265536034310890933, 11.15033473592831489620189273587, 11.93685012375342505576812726632

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