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Properties

Label 2-1815-5.4-c1-0-8
Degree $2$
Conductor $1815$
Sign $0.749 + 0.662i$
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  + 2.67i·2-s − i·3-s − 5.15·4-s + (−1.48 + 1.67i)5-s + 2.67·6-s + 2.80i·7-s − 8.44i·8-s − 9-s + (−4.48 − 3.96i)10-s + 5.15i·12-s + 5.11i·13-s − 7.50·14-s + (1.67 + 1.48i)15-s + 12.2·16-s + 4.54i·17-s − 2.67i·18-s + ⋯
L(s)  = 1  + 1.89i·2-s − 0.577i·3-s − 2.57·4-s + (−0.662 + 0.749i)5-s + 1.09·6-s + 1.06i·7-s − 2.98i·8-s − 0.333·9-s + (−1.41 − 1.25i)10-s + 1.48i·12-s + 1.41i·13-s − 2.00·14-s + (0.432 + 0.382i)15-s + 3.06·16-s + 1.10i·17-s − 0.630i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.749 + 0.662i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4522439479\)
\(L(\frac12)\) \(\approx\) \(0.4522439479\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 1.67i)T \)
11 \( 1 \)
good2 \( 1 - 2.67iT - 2T^{2} \)
7 \( 1 - 2.80iT - 7T^{2} \)
13 \( 1 - 5.11iT - 13T^{2} \)
17 \( 1 - 4.54iT - 17T^{2} \)
19 \( 1 + 4.57T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + 0.962T + 31T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 - 2.38T + 41T^{2} \)
43 \( 1 + 2.80iT - 43T^{2} \)
47 \( 1 - 4.31iT - 47T^{2} \)
53 \( 1 + 6.57iT - 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 7.92T + 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 6.41iT - 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 - 0.806iT - 83T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 - 9.92iT - 97T^{2} \)
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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

−9.390594421018851672159520172014, −8.840868014539593511921099397386, −8.159004404822910850148552487311, −7.50870981745546003413809500064, −6.70123729022110222498832350184, −6.26697563451826848977062489852, −5.53039501133645079398547404203, −4.41056493132618463362505080502, −3.63480428267262917058677541689, −2.04715633323752515776129872103, 0.20967433896678796843368782995, 1.01169863256713078039844389528, 2.56011796082708756456504158936, 3.45799172365067693690704123452, 4.20789378806500242954852338710, 4.73319813848862756907583063950, 5.60226967289747397931791577417, 7.31625283021032935276211877491, 8.234389210404950462178086693908, 8.798064919124597107195252151793

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