Properties

Label 2-385-7.2-c1-0-23
Degree $2$
Conductor $385$
Sign $0.140 + 0.990i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
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  + (1.02 − 1.78i)2-s + (0.831 + 1.43i)3-s + (−1.11 − 1.93i)4-s + (0.5 − 0.866i)5-s + 3.41·6-s + (−1.49 − 2.18i)7-s − 0.473·8-s + (0.118 − 0.205i)9-s + (−1.02 − 1.78i)10-s + (0.5 + 0.866i)11-s + (1.85 − 3.21i)12-s + 1.53·13-s + (−5.42 + 0.423i)14-s + 1.66·15-s + (1.74 − 3.01i)16-s + (0.944 + 1.63i)17-s + ⋯
L(s)  = 1  + (0.727 − 1.25i)2-s + (0.479 + 0.831i)3-s + (−0.557 − 0.965i)4-s + (0.223 − 0.387i)5-s + 1.39·6-s + (−0.565 − 0.824i)7-s − 0.167·8-s + (0.0395 − 0.0685i)9-s + (−0.325 − 0.563i)10-s + (0.150 + 0.261i)11-s + (0.535 − 0.926i)12-s + 0.426·13-s + (−1.44 + 0.113i)14-s + 0.429·15-s + (0.435 − 0.754i)16-s + (0.229 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.140 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75248 - 1.52101i\)
\(L(\frac12)\) \(\approx\) \(1.75248 - 1.52101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.49 + 2.18i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.02 + 1.78i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.831 - 1.43i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.54 - 2.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.476 - 0.825i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.17T + 29T^{2} \)
31 \( 1 + (-1.88 - 3.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 + 3.36T + 43T^{2} \)
47 \( 1 + (3.84 - 6.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.824 + 1.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.66 - 6.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.53 - 9.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.95 - 3.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.15T + 71T^{2} \)
73 \( 1 + (4.71 + 8.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-6.09 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−11.03122180786097971817666685319, −10.21447022848493100806515952829, −9.765061635802627340785283089044, −8.767349634961390943640304268611, −7.41771980224508378182494337460, −6.00270956722771014870034892752, −4.62234375770366056589774282033, −3.90243848543765671639066127142, −3.16062328707634574668443088275, −1.48669928513565131059884301445, 2.13067113931931005371662344439, 3.50750500604948932401433766234, 5.02750578111443479020951033801, 6.04222824666231022665459067233, 6.70561110926631037321819567865, 7.55048286146622308596478783041, 8.407532816546623438236943642482, 9.372285644951569000929799427389, 10.67705093101439228654370509792, 11.84255899377033716653561619658

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