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Properties

Label 2-555-185.184-c1-0-16
Degree $2$
Conductor $555$
Sign $-0.0453 + 0.998i$
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  − 2.46·2-s − i·3-s + 4.09·4-s + (−1.94 + 1.09i)5-s + 2.46i·6-s + 1.91i·7-s − 5.16·8-s − 9-s + (4.80 − 2.71i)10-s − 3.25·11-s − 4.09i·12-s + 0.160·13-s − 4.73i·14-s + (1.09 + 1.94i)15-s + 4.55·16-s + 1.85·17-s + ⋯
L(s)  = 1  − 1.74·2-s − 0.577i·3-s + 2.04·4-s + (−0.870 + 0.491i)5-s + 1.00i·6-s + 0.724i·7-s − 1.82·8-s − 0.333·9-s + (1.51 − 0.858i)10-s − 0.982·11-s − 1.18i·12-s + 0.0445·13-s − 1.26i·14-s + (0.283 + 0.502i)15-s + 1.13·16-s + 0.449·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0453 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.0453 + 0.998i$
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: \((2,\ 555,\ (\ :1/2),\ -0.0453 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224681 - 0.235122i\)
\(L(\frac12)\) \(\approx\) \(0.224681 - 0.235122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 - 1.09i)T \)
37 \( 1 + (5.15 + 3.22i)T \)
good2 \( 1 + 2.46T + 2T^{2} \)
7 \( 1 - 1.91iT - 7T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
13 \( 1 - 0.160T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + 3.64iT - 19T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + 4.15iT - 29T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
41 \( 1 - 0.101T + 41T^{2} \)
43 \( 1 - 0.344T + 43T^{2} \)
47 \( 1 + 8.46iT - 47T^{2} \)
53 \( 1 + 2.83iT - 53T^{2} \)
59 \( 1 + 9.33iT - 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 + 7.49iT - 67T^{2} \)
71 \( 1 - 2.80T + 71T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 + 8.03iT - 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 - 1.54iT - 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−10.59421742707940767343475838128, −9.531028033657870809115541397223, −8.603888880568766241772142654835, −8.037613482880369892187627624277, −7.24270723934768261544972345079, −6.56343154469592238244326906731, −5.19392041250541802468470792994, −3.14600792700879431046801664508, −2.16754603666867550650473715079, −0.38735561995575065315451311037, 1.11111370739693114325232952083, 2.96930637004434204696242965344, 4.26735812479807857610169536139, 5.55229920487838834792814493164, 7.07074419178604845826697300674, 7.70477439564643497041577085551, 8.423691562326795364720789927020, 9.182157486042693026761543233709, 10.17788905783624258943169403958, 10.63776783596073625048319846250

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