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Properties

Label 2-1110-111.68-c1-0-41
Degree $2$
Conductor $1110$
Sign $-0.212 + 0.977i$
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 + (1.55 − 0.770i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.551 − 1.64i)6-s + 4.38·7-s + (−0.707 − 0.707i)8-s + (1.81 − 2.39i)9-s − 1.00·10-s − 4.49·11-s + (−0.770 − 1.55i)12-s + (1.45 − 1.45i)13-s + (3.09 − 3.09i)14-s + (−1.64 − 0.551i)15-s − 1.00·16-s + (−1.09 − 1.09i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.895 − 0.445i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.225 − 0.670i)6-s + 1.65·7-s + (−0.250 − 0.250i)8-s + (0.603 − 0.797i)9-s − 0.316·10-s − 1.35·11-s + (−0.222 − 0.447i)12-s + (0.403 − 0.403i)13-s + (0.828 − 0.828i)14-s + (−0.423 − 0.142i)15-s − 0.250·16-s + (−0.264 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.212 + 0.977i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.212 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.033234454\)
\(L(\frac12)\) \(\approx\) \(3.033234454\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 0.770i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.167 - 6.08i)T \)
good7 \( 1 - 4.38T + 7T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 + (-1.45 + 1.45i)T - 13iT^{2} \)
17 \( 1 + (1.09 + 1.09i)T + 17iT^{2} \)
19 \( 1 + (0.737 - 0.737i)T - 19iT^{2} \)
23 \( 1 + (-1.56 - 1.56i)T + 23iT^{2} \)
29 \( 1 + (-3.68 + 3.68i)T - 29iT^{2} \)
31 \( 1 + (0.303 + 0.303i)T + 31iT^{2} \)
41 \( 1 - 1.99T + 41T^{2} \)
43 \( 1 + (-0.568 + 0.568i)T - 43iT^{2} \)
47 \( 1 - 8.83iT - 47T^{2} \)
53 \( 1 - 8.07iT - 53T^{2} \)
59 \( 1 + (9.81 + 9.81i)T + 59iT^{2} \)
61 \( 1 + (-0.496 - 0.496i)T + 61iT^{2} \)
67 \( 1 - 7.31iT - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (2.04 - 2.04i)T - 79iT^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (11.8 - 11.8i)T - 89iT^{2} \)
97 \( 1 + (0.766 - 0.766i)T - 97iT^{2} \)
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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

−9.595115071151584467093434849504, −8.580845784831919990202657490292, −7.989689747058384600257939802778, −7.48166494714674446594776110930, −6.09573465255983205075755751363, −4.97960631041032537820622299926, −4.41257875640343898979764934960, −3.14925645756340402372642251288, −2.21554838383609174369914217957, −1.12465716866738727423511347796, 1.94461481543065043082558470637, 2.93546663721581741270837264969, 4.13454225576796622749942008268, 4.77719425210534672894853154395, 5.56087791561696560848664785484, 7.02157326842232229987557072963, 7.66182526440778500546567647944, 8.413764620927058499261482793797, 8.791313663903277614822959193423, 10.21558316332897415237138421457

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