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Properties

Label 2-1815-5.4-c1-0-46
Degree $2$
Conductor $1815$
Sign $0.295 - 0.955i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
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  + 0.803i·2-s − i·3-s + 1.35·4-s + (2.13 + 0.660i)5-s + 0.803·6-s − 0.508i·7-s + 2.69i·8-s − 9-s + (−0.530 + 1.71i)10-s − 1.35i·12-s + 5.13i·13-s + 0.408·14-s + (0.660 − 2.13i)15-s + 0.544·16-s + 3.34i·17-s − 0.803i·18-s + ⋯
L(s)  = 1  + 0.568i·2-s − 0.577i·3-s + 0.677·4-s + (0.955 + 0.295i)5-s + 0.327·6-s − 0.192i·7-s + 0.952i·8-s − 0.333·9-s + (−0.167 + 0.542i)10-s − 0.391i·12-s + 1.42i·13-s + 0.109·14-s + (0.170 − 0.551i)15-s + 0.136·16-s + 0.811i·17-s − 0.189i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.440035552\)
\(L(\frac12)\) \(\approx\) \(2.440035552\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 - 0.660i)T \)
11 \( 1 \)
good2 \( 1 - 0.803iT - 2T^{2} \)
7 \( 1 + 0.508iT - 7T^{2} \)
13 \( 1 - 5.13iT - 13T^{2} \)
17 \( 1 - 3.34iT - 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 - 2.91iT - 23T^{2} \)
29 \( 1 + 0.392T + 29T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 + 4.45iT - 37T^{2} \)
41 \( 1 - 2.79T + 41T^{2} \)
43 \( 1 - 3.05iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 9.80iT - 53T^{2} \)
59 \( 1 - 3.02T + 59T^{2} \)
61 \( 1 + 1.05T + 61T^{2} \)
67 \( 1 + 5.31iT - 67T^{2} \)
71 \( 1 - 4.09T + 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 - 9.84T + 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−9.254108980880308126410583609864, −8.608648596512075694570659222635, −7.47391372390776133561671256098, −7.10806132505676362433535839404, −6.12589009165974878198667845786, −5.93350099765408740649091103954, −4.71108132309812014457540579952, −3.39046131081474107672118081059, −2.14769375530631752924091698870, −1.64707953011926431327186923222, 0.874449920667772115228600001227, 2.25773606376065517178523454633, 2.89436433324864304388792447184, 3.93910271468968752338447299875, 5.18831938507882209879278214319, 5.69838619873218219648294110330, 6.62594072116450962204002871027, 7.51373111169033394670158477575, 8.569565790823138203610055652214, 9.269564246801398703998142654017

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