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Properties

Label 1-3300-3300.791-r1-0-0
Degree $1$
Conductor $3300$
Sign $-0.187 + 0.982i$
Analytic cond. $354.634$
Root an. cond. $354.634$
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  + 7-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.309 − 0.951i)37-s + (0.309 − 0.951i)41-s + 43-s + (−0.809 + 0.587i)47-s + 49-s + (0.809 − 0.587i)53-s + (0.309 − 0.951i)59-s + (−0.309 − 0.951i)61-s + ⋯
L(s)  = 1  + 7-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.309 − 0.951i)37-s + (0.309 − 0.951i)41-s + 43-s + (−0.809 + 0.587i)47-s + 49-s + (0.809 − 0.587i)53-s + (0.309 − 0.951i)59-s + (−0.309 − 0.951i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(354.634\)
Root analytic conductor: \(354.634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3300} (791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3300,\ (1:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027336002 + 1.241835901i\)
\(L(\frac12)\) \(\approx\) \(1.027336002 + 1.241835901i\)
\(L(1)\) \(\approx\) \(1.073568333 + 0.1046709909i\)
\(L(1)\) \(\approx\) \(1.073568333 + 0.1046709909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−18.42871664705371605606840380100, −17.79126810334921064846918616193, −17.13220993398217307648500027123, −16.64449584850091484044077642483, −15.533807893601644374464191875170, −14.89052159896739309683184037507, −14.65063792288626221585070647076, −13.484480689455662424098743295643, −13.02633079814925856829585803668, −12.164744137684316707679615131162, −11.47634002954412384843525046575, −10.68366946179708910014012408352, −10.24926386632093831235452219519, −9.23371652063191490573017878718, −8.29437253000199552551693202440, −8.06044541916681261143293598146, −7.09241965033531585253491151454, −6.167931162354997545642061252, −5.55289555362389120184417580978, −4.49954518049481925367686751376, −4.18121882140265433306274318537, −2.88012780378236210781379839754, −2.19866398933627168584044149672, −1.27519038919625110493830238837, −0.27732473732649364309726672578, 0.86314086484362035323628983628, 1.90872195924428728696655476, 2.41745573970166510412405746561, 3.63313892990436820507655881684, 4.478589432933390286085052147706, 4.984802629198058048135576598314, 5.87561607781677939411987603665, 6.91431779276154492197015272459, 7.318574554246349118708542907011, 8.30033201047531029437735419894, 9.0042762542974919247470855288, 9.52050147480302328748505775194, 10.66358343735583258186759277042, 11.21486836680770805639294131254, 11.70732577923779101499120437580, 12.62861571397971186230577800632, 13.35774937232378166207019051202, 14.178368628280277890187315692061, 14.57509473060216619012108717327, 15.48105336932481403181604174914, 16.024363634879479195022803374484, 16.986123583913572516813820636, 17.54158035893754101585695298668, 18.03837671125676445590146592312, 18.98782373683278797661114322490

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