Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.626 + 0.779i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.13i·3-s + 3.91i·5-s − 2.98i·7-s − 6.81·9-s + 5.81i·11-s + 1.72·13-s − 12.2·15-s + (−2.58 − 3.21i)17-s − 0.658·19-s + 9.34·21-s − 9.22i·23-s − 10.3·25-s − 11.9i·27-s − 10.2i·29-s + 5.64i·31-s + ⋯
L(s)  = 1  + 1.80i·3-s + 1.75i·5-s − 1.12i·7-s − 2.27·9-s + 1.75i·11-s + 0.477·13-s − 3.16·15-s + (−0.626 − 0.779i)17-s − 0.151·19-s + 2.03·21-s − 1.92i·23-s − 2.06·25-s − 2.29i·27-s − 1.89i·29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.626 + 0.779i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.626 + 0.779i)$
$L(1)$  $\approx$  $0.1866376737$
$L(\frac12)$  $\approx$  $0.1866376737$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (2.58 + 3.21i)T \)
59 \( 1 - T \)
good3 \( 1 - 3.13iT - 3T^{2} \)
5 \( 1 - 3.91iT - 5T^{2} \)
7 \( 1 + 2.98iT - 7T^{2} \)
11 \( 1 - 5.81iT - 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
19 \( 1 + 0.658T + 19T^{2} \)
23 \( 1 + 9.22iT - 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 - 5.64iT - 31T^{2} \)
37 \( 1 - 3.51iT - 37T^{2} \)
41 \( 1 + 0.336iT - 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + 4.71T + 47T^{2} \)
53 \( 1 + 3.06T + 53T^{2} \)
61 \( 1 + 4.36iT - 61T^{2} \)
67 \( 1 - 4.44T + 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 3.72iT - 73T^{2} \)
79 \( 1 + 5.49iT - 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 9.45T + 89T^{2} \)
97 \( 1 - 9.73iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.445579652295108878490919389112, −8.393421311301739248893530760012, −7.53988500336908293427027424246, −6.71348645249315150857904490588, −6.37116160741060816258630065159, −5.00378381320817764089953227544, −4.36677525439476819730649059288, −3.92028791018785538504147536593, −2.97540378225554353888262065050, −2.29003341511536415118518949076, 0.05310055410094804374390805282, 1.25279261467420785202388675120, 1.67230273496818074355762029901, 2.85328328529584026544973996980, 3.84602935094540226736159641760, 5.32687638121387707511874996646, 5.64880962120692947468419047548, 6.14361054127729133612364066042, 7.13979246809194214028084829712, 8.115471097393173461100548459069

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