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Properties

Label 2-1610-805.804-c1-0-68
Degree $2$
Conductor $1610$
Sign $0.953 + 0.299i$
Analytic cond. $12.8559$
Root an. cond. $3.58551$
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  − i·2-s + 3.32·3-s − 4-s + (−0.0164 + 2.23i)5-s − 3.32i·6-s + (2.20 − 1.45i)7-s + i·8-s + 8.02·9-s + (2.23 + 0.0164i)10-s + 3.09i·11-s − 3.32·12-s + 3.13·13-s + (−1.45 − 2.20i)14-s + (−0.0547 + 7.42i)15-s + 16-s + 0.517i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.91·3-s − 0.5·4-s + (−0.00736 + 0.999i)5-s − 1.35i·6-s + (0.834 − 0.550i)7-s + 0.353i·8-s + 2.67·9-s + (0.707 + 0.00520i)10-s + 0.933i·11-s − 0.958·12-s + 0.868·13-s + (−0.389 − 0.590i)14-s + (−0.0141 + 1.91i)15-s + 0.250·16-s + 0.125i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1610\)    =    \(2 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.953 + 0.299i$
Analytic conductor: \(12.8559\)
Root analytic conductor: \(3.58551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1610} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1610,\ (\ :1/2),\ 0.953 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.612050524\)
\(L(\frac12)\) \(\approx\) \(3.612050524\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.0164 - 2.23i)T \)
7 \( 1 + (-2.20 + 1.45i)T \)
23 \( 1 + (3.05 + 3.69i)T \)
good3 \( 1 - 3.32T + 3T^{2} \)
11 \( 1 - 3.09iT - 11T^{2} \)
13 \( 1 - 3.13T + 13T^{2} \)
17 \( 1 - 0.517iT - 17T^{2} \)
19 \( 1 + 4.30T + 19T^{2} \)
29 \( 1 + 7.27T + 29T^{2} \)
31 \( 1 - 3.31iT - 31T^{2} \)
37 \( 1 - 6.17T + 37T^{2} \)
41 \( 1 - 5.26iT - 41T^{2} \)
43 \( 1 + 1.14T + 43T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 3.04T + 67T^{2} \)
71 \( 1 + 3.83T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 + 1.67iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 - 2.13T + 89T^{2} \)
97 \( 1 - 6.62iT - 97T^{2} \)
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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

−9.502235602721397926436829486196, −8.445210130110486587505114598768, −8.035990457228503017411987798707, −7.27063447742752006288294483653, −6.41702216785807665889503529507, −4.64029991895327896587112048797, −3.98721364213540530813783473054, −3.29025705369210556791201255701, −2.20762950106148250077586260975, −1.69436360690183309838818150263, 1.33852048020677537418561844652, 2.32676057001654863780986903126, 3.69880625002692099373778231189, 4.20296680508503917027754200542, 5.32235073953353866362824961045, 6.19516595610861665573306425194, 7.51503282827415559609625733798, 8.107737472927293949753096100470, 8.493626291062589855858876988714, 9.114217274352850419127480232146

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