Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.727 - 0.686i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.79i·3-s + 1.19i·5-s − 1.20i·7-s − 0.216·9-s − 0.564i·11-s − 6.78·13-s + 2.15·15-s + (−2.99 + 2.83i)17-s − 5.87·19-s − 2.16·21-s − 2.74i·23-s + 3.56·25-s − 4.99i·27-s + 8.94i·29-s − 4.79i·31-s + ⋯
L(s)  = 1  − 1.03i·3-s + 0.536i·5-s − 0.455i·7-s − 0.0720·9-s − 0.170i·11-s − 1.88·13-s + 0.555·15-s + (−0.727 + 0.686i)17-s − 1.34·19-s − 0.471·21-s − 0.571i·23-s + 0.712·25-s − 0.960i·27-s + 1.66i·29-s − 0.862i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.727 - 0.686i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.727 - 0.686i)$
$L(1)$  $\approx$  $1.047937593$
$L(\frac12)$  $\approx$  $1.047937593$
$L(\frac{3}{2})$   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 \( 1 + (2.99 - 2.83i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.79iT - 3T^{2} \)
5 \( 1 - 1.19iT - 5T^{2} \)
7 \( 1 + 1.20iT - 7T^{2} \)
11 \( 1 + 0.564iT - 11T^{2} \)
13 \( 1 + 6.78T + 13T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 + 2.74iT - 23T^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 + 4.79iT - 31T^{2} \)
37 \( 1 - 5.32iT - 37T^{2} \)
41 \( 1 - 6.25iT - 41T^{2} \)
43 \( 1 - 8.95T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
61 \( 1 + 0.445iT - 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + 9.22iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 + 6.77T + 83T^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−8.407236785260992410922741439337, −7.57879587794586695496190094685, −7.00772730958273234538274635703, −6.66456273498949762735270734094, −5.77490518868662415979442895987, −4.65130487047085378301039617949, −4.07047837292203533066870655366, −2.65582775009713280413160698226, −2.25087304286201605123492366689, −0.988239257726855629426764499448, 0.33469224336114838642052407938, 2.10632246732894112635861748247, 2.73747982384821717327624266672, 4.16526453440980192392863159380, 4.43543206928410080029752330566, 5.21888865503205583664634647668, 5.88603258856100440656196314672, 7.13598329850461831076003759355, 7.45345139361523207207814822059, 8.850532582899476912970414685280

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