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Properties

Label 2-1815-5.4-c1-0-103
Degree $2$
Conductor $1815$
Sign $-0.841 - 0.540i$
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  − 1.53i·2-s − i·3-s − 0.361·4-s + (1.20 − 1.88i)5-s − 1.53·6-s − 2.86i·7-s − 2.51i·8-s − 9-s + (−2.89 − 1.85i)10-s + 0.361i·12-s − 2.43i·13-s − 4.40·14-s + (−1.88 − 1.20i)15-s − 4.59·16-s − 3.88i·17-s + 1.53i·18-s + ⋯
L(s)  = 1  − 1.08i·2-s − 0.577i·3-s − 0.180·4-s + (0.540 − 0.841i)5-s − 0.627·6-s − 1.08i·7-s − 0.890i·8-s − 0.333·9-s + (−0.914 − 0.587i)10-s + 0.104i·12-s − 0.675i·13-s − 1.17·14-s + (−0.485 − 0.311i)15-s − 1.14·16-s − 0.942i·17-s + 0.362i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.841 - 0.540i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.102834323\)
\(L(\frac12)\) \(\approx\) \(2.102834323\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.88i)T \)
11 \( 1 \)
good2 \( 1 + 1.53iT - 2T^{2} \)
7 \( 1 + 2.86iT - 7T^{2} \)
13 \( 1 + 2.43iT - 13T^{2} \)
17 \( 1 + 3.88iT - 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 - 6.02iT - 23T^{2} \)
29 \( 1 - 4.03T + 29T^{2} \)
31 \( 1 + 0.783T + 31T^{2} \)
37 \( 1 - 8.02iT - 37T^{2} \)
41 \( 1 - 7.54T + 41T^{2} \)
43 \( 1 - 9.90iT - 43T^{2} \)
47 \( 1 - 0.551iT - 47T^{2} \)
53 \( 1 + 2.75iT - 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 - 3.96iT - 67T^{2} \)
71 \( 1 - 9.27T + 71T^{2} \)
73 \( 1 - 0.615iT - 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 8.95iT - 97T^{2} \)
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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

−9.097257393388234982818203688408, −7.900698901354944090888514700972, −7.36069343415449231444881156168, −6.45419486501155228760376109577, −5.43254764404431512287284974506, −4.56520590136589435764693711961, −3.44782985970134246224240652900, −2.64145348021814313126417858167, −1.31259597059321063871959919540, −0.847621137827232349408458084751, 2.08983078715098855703841046951, 2.80988174081740452811588036846, 4.07682287964041685363511337085, 5.29800061315355490019403645159, 5.75978560214531392630653792651, 6.49216919809732691806417555220, 7.18356080291845501069887140996, 8.135248329633958380739361446845, 8.907242174101396924480027698497, 9.474476840449884454717481325305

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