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Properties

Label 2-3332-476.411-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.347 + 0.937i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.128 − 1.95i)5-s + (0.382 − 0.923i)8-s + (0.793 − 0.608i)9-s + (1.95 + 0.128i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (−0.793 − 0.608i)17-s + (0.499 + 0.866i)18-s + (−0.382 + 1.92i)20-s + (−2.82 + 0.371i)25-s + (0.860 + 1.12i)26-s + (0.324 − 0.216i)29-s + (−0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.128 − 1.95i)5-s + (0.382 − 0.923i)8-s + (0.793 − 0.608i)9-s + (1.95 + 0.128i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (−0.793 − 0.608i)17-s + (0.499 + 0.866i)18-s + (−0.382 + 1.92i)20-s + (−2.82 + 0.371i)25-s + (0.860 + 1.12i)26-s + (0.324 − 0.216i)29-s + (−0.608 + 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.347 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9912457867\)
\(L(\frac12)\) \(\approx\) \(0.9912457867\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 - 0.991i)T \)
7 \( 1 \)
17 \( 1 + (0.793 + 0.608i)T \)
good3 \( 1 + (-0.793 + 0.608i)T^{2} \)
5 \( 1 + (0.128 + 1.95i)T + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (-0.130 + 0.991i)T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + (-0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.793 + 0.608i)T^{2} \)
29 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.793 - 0.608i)T^{2} \)
37 \( 1 + (1.29 - 1.47i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.735 - 1.49i)T + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.349 + 0.172i)T + (0.608 + 0.793i)T^{2} \)
79 \( 1 + (-0.793 - 0.608i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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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

−8.728676690915642826902871405558, −8.062471248254033153252278119727, −7.24202283928280806683996924779, −6.38664147029516093054240544054, −5.57792977077306406378640655485, −4.93641175348457087717903501684, −4.29376998649282583500415983684, −3.52056645916355733395876619363, −1.50758829616401443202133940713, −0.66786727119881126711942151945, 1.79750003065891144287353001632, 2.30866672732066112728074324176, 3.53235781674946339323971651221, 3.85276675974790049312518160346, 4.88078585404604389826321261404, 6.11124090368320196409492683482, 6.79469199251326805906791774600, 7.42924270959493464038606678185, 8.302319822885855883034907232817, 9.060400196338761016817107315196

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