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Properties

Label 2-3536-884.155-c0-0-1
Degree $2$
Conductor $3536$
Sign $0.673 + 0.739i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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  + (0.292 − 0.707i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s − i·13-s + 17-s + (−1 − i)19-s + (−0.707 + 0.707i)25-s + (−0.707 − 1.70i)29-s + (−1.70 + 0.707i)31-s + 2i·47-s + (0.292 + 0.292i)49-s + (−1 + i)59-s + (0.292 − 0.707i)61-s + (−0.292 − 0.707i)63-s + 1.41·67-s + ⋯
L(s)  = 1  + (0.292 − 0.707i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s − i·13-s + 17-s + (−1 − i)19-s + (−0.707 + 0.707i)25-s + (−0.707 − 1.70i)29-s + (−1.70 + 0.707i)31-s + 2i·47-s + (0.292 + 0.292i)49-s + (−1 + i)59-s + (0.292 − 0.707i)61-s + (−0.292 − 0.707i)63-s + 1.41·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.499277176\)
\(L(\frac12)\) \(\approx\) \(1.499277176\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
17 \( 1 - T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (1 - i)T - iT^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−8.775801218066058847178694764028, −7.59814388904729068551794702637, −7.35409264639173351513302463897, −6.45021749863520625541663044392, −5.78441157216822722204420176419, −4.61468409084732001912786750308, −4.01296245533721836591930632368, −3.36890111900793522041258081456, −1.89165184224154220249585687522, −0.987063898612726495354128234983, 1.56631331079462135252561265614, 2.05049512282581678332565981703, 3.71573413621836351189677079384, 3.95331049723018105715266706884, 5.16986754801424293037742594531, 5.82554737287717930316175855053, 6.64628932619176871642663951621, 7.29820354540385725768237394025, 8.266054626820765042242427050347, 8.790735454313308133507810841983

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