Properties

Degree 2
Conductor $ 2^{2} \cdot 17 $
Sign $0.615 + 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + (−1 + i)5-s + i·8-s i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s i·25-s + (1 − i)29-s i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯
L(s)  = 1  i·2-s − 4-s + (−1 + i)5-s + i·8-s i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s i·25-s + (1 − i)29-s i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $0.615 + 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{68} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 68,\ (\ :0),\ 0.615 + 0.788i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4210945963\)
\(L(\frac12)\)  \(\approx\)  \(0.4210945963\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + iT \)
17 \( 1 + T \)
good3 \( 1 + iT^{2} \)
5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−14.84625762244638749011823645362, −13.73708928432188607873060131214, −12.36784286101715037183258870678, −11.54553781733229543264173850657, −10.67475820275141773886786578035, −9.401820822599483704682833492791, −8.063910698916515610767798864707, −6.52495137986341477590907454457, −4.29878395807902601847606510864, −3.03544157255568814852703229915, 4.22564061217201658866752193569, 5.27019738107696468255483049268, 7.09193503008472694559106097842, 8.208046929617629467128032615792, 8.986147440681671677679816455417, 10.68372419403733204059333291826, 12.21051665090741868031115105838, 13.15884627493339383429814612152, 14.21237496700354657276706814216, 15.59461919004984204952321323569

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