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Properties

Label 2-440-40.29-c1-0-23
Degree $2$
Conductor $440$
Sign $-0.427 - 0.904i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  + (0.109 + 1.40i)2-s + 2.58·3-s + (−1.97 + 0.308i)4-s + (0.464 + 2.18i)5-s + (0.283 + 3.65i)6-s + 1.54i·7-s + (−0.650 − 2.75i)8-s + 3.70·9-s + (−3.03 + 0.894i)10-s + i·11-s + (−5.11 + 0.798i)12-s − 0.752·13-s + (−2.18 + 0.169i)14-s + (1.20 + 5.66i)15-s + (3.80 − 1.21i)16-s − 1.76i·17-s + ⋯
L(s)  = 1  + (0.0772 + 0.997i)2-s + 1.49·3-s + (−0.988 + 0.154i)4-s + (0.207 + 0.978i)5-s + (0.115 + 1.49i)6-s + 0.585i·7-s + (−0.230 − 0.973i)8-s + 1.23·9-s + (−0.959 + 0.282i)10-s + 0.301i·11-s + (−1.47 + 0.230i)12-s − 0.208·13-s + (−0.583 + 0.0452i)14-s + (0.310 + 1.46i)15-s + (0.952 − 0.304i)16-s − 0.427i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12834 + 1.78115i\)
\(L(\frac12)\) \(\approx\) \(1.12834 + 1.78115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.109 - 1.40i)T \)
5 \( 1 + (-0.464 - 2.18i)T \)
11 \( 1 - iT \)
good3 \( 1 - 2.58T + 3T^{2} \)
7 \( 1 - 1.54iT - 7T^{2} \)
13 \( 1 + 0.752T + 13T^{2} \)
17 \( 1 + 1.76iT - 17T^{2} \)
19 \( 1 + 6.61iT - 19T^{2} \)
23 \( 1 - 4.37iT - 23T^{2} \)
29 \( 1 + 1.83iT - 29T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 - 6.83T + 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 + 1.78iT - 47T^{2} \)
53 \( 1 + 3.54T + 53T^{2} \)
59 \( 1 - 0.926iT - 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 3.66iT - 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 15.4iT - 97T^{2} \)
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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

−11.40180324704795646615921360762, −10.01166748092444075803361935990, −9.348550244185809939060138487218, −8.668059063493717166098815523872, −7.61222367469305851028029455706, −7.07736689708357845872287340490, −5.94634591723889458352853551339, −4.62361168732094955582416641420, −3.32697423127235013967256531248, −2.43920394212800001836567171162, 1.30050565760211578945682205739, 2.53422351740276024202945262787, 3.75705620070136771244273191709, 4.47241833984667646842303731633, 5.86641742219444669558743997741, 7.69446122744948270946089093097, 8.414328425862950519367818595828, 9.005537488322462686472235541164, 9.931691978990723933569288628947, 10.54360925434893724736591561109

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