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Properties

Label 2-1110-111.68-c1-0-0
Degree $2$
Conductor $1110$
Sign $-0.998 + 0.0591i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (0.707 − 0.707i)2-s + (−0.544 + 1.64i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (0.778 + 1.54i)6-s − 1.56·7-s + (−0.707 − 0.707i)8-s + (−2.40 − 1.78i)9-s + 1.00·10-s − 1.64·11-s + (1.64 + 0.544i)12-s + (−0.629 + 0.629i)13-s + (−1.10 + 1.10i)14-s + (−1.54 + 0.778i)15-s − 1.00·16-s + (−4.46 − 4.46i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.314 + 0.949i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.317 + 0.631i)6-s − 0.592·7-s + (−0.250 − 0.250i)8-s + (−0.802 − 0.596i)9-s + 0.316·10-s − 0.495·11-s + (0.474 + 0.157i)12-s + (−0.174 + 0.174i)13-s + (−0.296 + 0.296i)14-s + (−0.399 + 0.200i)15-s − 0.250·16-s + (−1.08 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.998 + 0.0591i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.998 + 0.0591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03768993140\)
\(L(\frac12)\) \(\approx\) \(0.03768993140\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.544 - 1.64i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (5.36 + 2.85i)T \)
good7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 1.64T + 11T^{2} \)
13 \( 1 + (0.629 - 0.629i)T - 13iT^{2} \)
17 \( 1 + (4.46 + 4.46i)T + 17iT^{2} \)
19 \( 1 + (4.49 - 4.49i)T - 19iT^{2} \)
23 \( 1 + (-1.08 - 1.08i)T + 23iT^{2} \)
29 \( 1 + (0.289 - 0.289i)T - 29iT^{2} \)
31 \( 1 + (5.62 + 5.62i)T + 31iT^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (0.834 - 0.834i)T - 43iT^{2} \)
47 \( 1 - 13.1iT - 47T^{2} \)
53 \( 1 + 4.14iT - 53T^{2} \)
59 \( 1 + (8.55 + 8.55i)T + 59iT^{2} \)
61 \( 1 + (9.70 + 9.70i)T + 61iT^{2} \)
67 \( 1 - 4.75iT - 67T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (0.689 - 0.689i)T - 79iT^{2} \)
83 \( 1 + 2.00iT - 83T^{2} \)
89 \( 1 + (-7.48 + 7.48i)T - 89iT^{2} \)
97 \( 1 + (-3.88 + 3.88i)T - 97iT^{2} \)
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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

−10.43916017503030808720840768935, −9.429674544565382494151392333827, −9.206943247639483933631322955876, −7.77527324142448969433638783953, −6.54786530011904544532915002747, −5.92952422830310819705545554415, −4.97315511076688655313074376584, −4.14592505295948161075712358120, −3.18805974247781006494203333719, −2.21687620354568953299852468775, 0.01313373029694215274462569054, 1.92586151022702166200310223325, 2.96084855062030355578357511151, 4.37510117167360692805953779299, 5.30063604054339544985876406378, 6.17216912733703463864050333736, 6.74487210551782704906928322572, 7.53687796767824870512400241284, 8.589426891619885608695149138028, 9.021863480357539792826675804743

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