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Properties

Label 2-385-7.4-c1-0-27
Degree $2$
Conductor $385$
Sign $-0.981 + 0.190i$
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  + (−0.420 − 0.727i)2-s + (0.864 − 1.49i)3-s + (0.647 − 1.12i)4-s + (−0.5 − 0.866i)5-s − 1.45·6-s + (−2.55 − 0.702i)7-s − 2.76·8-s + (0.00412 + 0.00713i)9-s + (−0.420 + 0.727i)10-s + (0.5 − 0.866i)11-s + (−1.11 − 1.93i)12-s − 2.36·13-s + (0.560 + 2.15i)14-s − 1.72·15-s + (−0.132 − 0.228i)16-s + (1.34 − 2.32i)17-s + ⋯
L(s)  = 1  + (−0.296 − 0.514i)2-s + (0.499 − 0.864i)3-s + (0.323 − 0.560i)4-s + (−0.223 − 0.387i)5-s − 0.593·6-s + (−0.964 − 0.265i)7-s − 0.978·8-s + (0.00137 + 0.00237i)9-s + (−0.132 + 0.230i)10-s + (0.150 − 0.261i)11-s + (−0.323 − 0.559i)12-s − 0.655·13-s + (0.149 + 0.574i)14-s − 0.446·15-s + (−0.0330 − 0.0571i)16-s + (0.326 − 0.564i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.981 + 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108316 - 1.12819i\)
\(L(\frac12)\) \(\approx\) \(0.108316 - 1.12819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 + 0.702i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.420 + 0.727i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.864 + 1.49i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 2.36T + 13T^{2} \)
17 \( 1 + (-1.34 + 2.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.425 - 0.737i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.71 - 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 + (-4.32 + 7.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.26 + 9.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.73T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 + (-0.618 - 1.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.793 - 1.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.859 + 1.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.78 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.37 - 2.37i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 + (-4.18 + 7.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + (-5.66 - 9.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.868T + 97T^{2} \)
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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

−10.84686018661313127358572099519, −9.914472158292110187527067045748, −9.260260392404180430038394647064, −8.126299696237799349612603874134, −7.13699103585417975707898919153, −6.35126695151004422776707636986, −5.06359390598985690572820134641, −3.32861117463678988836782398219, −2.23321543283650996373337322635, −0.77443283189413134037851993184, 2.85069028978474951132969605827, 3.46353219191554743960651916525, 4.80081003917142372765001772696, 6.49303368630305764894565887458, 6.89393112288183645888597730344, 8.262973855980769410978287447796, 8.881083381379786538042423900011, 9.922720776902751447341622482076, 10.45003436607886221608756032770, 12.10503280243527105809595276256

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