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Properties

Label 2-3377-3377.306-c0-0-1
Degree $2$
Conductor $3377$
Sign $0.998 + 0.0589i$
Analytic cond. $1.68534$
Root an. cond. $1.29820$
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  + (−0.809 − 0.587i)4-s + (1.58 + 1.14i)7-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)11-s + (0.309 + 0.951i)16-s + (0.564 + 1.73i)17-s + (0.169 − 0.122i)19-s + (−0.809 + 0.587i)25-s + (−0.604 − 1.86i)28-s + (−0.809 + 0.587i)36-s + (−1.08 − 0.786i)37-s + (0.809 − 0.587i)41-s + (−0.978 + 0.207i)44-s + (0.873 + 2.68i)49-s + (−0.309 + 0.951i)53-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (1.58 + 1.14i)7-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)11-s + (0.309 + 0.951i)16-s + (0.564 + 1.73i)17-s + (0.169 − 0.122i)19-s + (−0.809 + 0.587i)25-s + (−0.604 − 1.86i)28-s + (−0.809 + 0.587i)36-s + (−1.08 − 0.786i)37-s + (0.809 − 0.587i)41-s + (−0.978 + 0.207i)44-s + (0.873 + 2.68i)49-s + (−0.309 + 0.951i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3377 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3377 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3377\)    =    \(11 \cdot 307\)
Sign: $0.998 + 0.0589i$
Analytic conductor: \(1.68534\)
Root analytic conductor: \(1.29820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3377} (306, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3377,\ (\ :0),\ 0.998 + 0.0589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361913562\)
\(L(\frac12)\) \(\approx\) \(1.361913562\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.669 + 0.743i)T \)
307 \( 1 - T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−8.936862668672153103157553655525, −8.260935273401168133549491563973, −7.49521463612813877726497855883, −6.03817888109875169418281504260, −5.93166315921948819451654756073, −5.05410643157842871010907272931, −4.14738101025874326341990164878, −3.50476364250161628093924348691, −1.90389611744823922741826482957, −1.22932055045152719247694834448, 1.08992262482100934275790163784, 2.15932795386439778014920500053, 3.48125675453903460801769978816, 4.44995896943508567822988242834, 4.70442960033689467289662701214, 5.41342797022708109931154616829, 7.00434285336977010498202197964, 7.41549426473290208620310108177, 8.003013288550039588254195092554, 8.569952720531577940232441164098

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