Properties

Label 2-555-185.184-c1-0-17
Degree $2$
Conductor $555$
Sign $0.774 - 0.632i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  − 2.71·2-s + i·3-s + 5.37·4-s + (0.734 + 2.11i)5-s − 2.71i·6-s − 1.35i·7-s − 9.16·8-s − 9-s + (−1.99 − 5.73i)10-s + 2.97·11-s + 5.37i·12-s + 5.29·13-s + 3.68i·14-s + (−2.11 + 0.734i)15-s + 14.1·16-s + 5.19·17-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577i·3-s + 2.68·4-s + (0.328 + 0.944i)5-s − 1.10i·6-s − 0.513i·7-s − 3.23·8-s − 0.333·9-s + (−0.630 − 1.81i)10-s + 0.898·11-s + 1.55i·12-s + 1.46·13-s + 0.985i·14-s + (−0.545 + 0.189i)15-s + 3.53·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702560 + 0.250263i\)
\(L(\frac12)\) \(\approx\) \(0.702560 + 0.250263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (-0.734 - 2.11i)T \)
37 \( 1 + (-5.71 + 2.08i)T \)
good2 \( 1 + 2.71T + 2T^{2} \)
7 \( 1 + 1.35iT - 7T^{2} \)
11 \( 1 - 2.97T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 + 8.16iT - 19T^{2} \)
23 \( 1 + 2.36T + 23T^{2} \)
29 \( 1 + 8.38iT - 29T^{2} \)
31 \( 1 - 6.98iT - 31T^{2} \)
41 \( 1 - 0.765T + 41T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 - 2.48iT - 47T^{2} \)
53 \( 1 + 3.57iT - 53T^{2} \)
59 \( 1 + 0.162iT - 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 2.92iT - 67T^{2} \)
71 \( 1 + 7.77T + 71T^{2} \)
73 \( 1 - 1.81iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 - 6.27iT - 83T^{2} \)
89 \( 1 - 7.15iT - 89T^{2} \)
97 \( 1 + 9.38T + 97T^{2} \)
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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

−10.65053458515107596697418165716, −9.921519764067856412322087727177, −9.262142764534465007021884346101, −8.433129096278270266964246752567, −7.44376255000035887895873241290, −6.62195278553427941700320427352, −5.89870925782799351682199093753, −3.76450257766690586997383805353, −2.66628462010546762545657942745, −1.09589951545934872337016563805, 1.13509380186302977391172677859, 1.78302233873464517471270156757, 3.51200062948679100331335859045, 5.88302619848480625254577657579, 6.12171904971973177769900619229, 7.53187583864224489520973412255, 8.273320355245526757597134927794, 8.803501185326057629270080493595, 9.604344823596044781693034124134, 10.36696808101456339411812415183

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