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Properties

Label 1-385-385.349-r0-0-0
Degree $1$
Conductor $385$
Sign $0.530 + 0.847i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s − 23-s + (0.309 − 0.951i)24-s + (0.809 + 0.587i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s − 23-s + (0.309 − 0.951i)24-s + (0.809 + 0.587i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4991873235 + 0.2766128210i\)
\(L(\frac12)\) \(\approx\) \(0.4991873235 + 0.2766128210i\)
\(L(1)\) \(\approx\) \(0.7115302230 - 0.1211801429i\)
\(L(1)\) \(\approx\) \(0.7115302230 - 0.1211801429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.13327794281106423975691451735, −23.73498189453830948159131438104, −22.71530941600994770761541098420, −22.13027310944719687309945935657, −21.293700185415217733966834873482, −19.79957237971929294568810164683, −18.87186800350656935900415305142, −17.77016133375385728770976403474, −17.43855484779923442630683853201, −16.47866694962027540554475292552, −15.56309348874438972855189025997, −14.71913973709518709345989867057, −13.51606117376771879870241757714, −12.86207049293864471825835830589, −12.0955747146649008763049709628, −10.88889982255689910327790622573, −9.82702436970958976582998290069, −8.36958083079085243092359639836, −7.73070329918798397645677991472, −6.59570064378354611188821488709, −5.94320405517603140961103246690, −4.96846504746079680460609977510, −3.925367469578848677917121240639, −2.26422708389347424198495952657, −0.356389780231627068750185043235, 1.38523644023379621113564749382, 2.79096604878814452044871715299, 4.10225588674668528316518791027, 4.71724297704333465753091566361, 5.817756951765343599571797468752, 6.81892621077621199843015335522, 8.571264812677002907716183973973, 9.55978514436661159781887173511, 10.27211592114099158887829170670, 11.23097180185303978358561281914, 11.93295518841234241851159421886, 12.68989673883424607719701762607, 13.95167416418249128594865590884, 14.699106699118261689589366085098, 15.869795910058161348971669331694, 16.733335212483641120280572588089, 17.816835872678331372248834296322, 18.47090323001265793131933805587, 19.55170773337861226382172998133, 20.468110516003255853657398223743, 21.34648260674800095848158550479, 21.89892451873577512348721951890, 22.73792468456177417509282206195, 23.53388061722610129626443685773, 24.19211194576848793706457765943

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