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Properties

Label 2-51e2-17.14-c0-0-0
Degree $2$
Conductor $2601$
Sign $0.764 - 0.644i$
Analytic cond. $1.29806$
Root an. cond. $1.13932$
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  + (−0.707 + 0.707i)4-s + (−1.44 − 0.962i)7-s + (0.707 + 0.707i)13-s − 1.00i·16-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)25-s + (1.69 − 0.337i)28-s + (1.69 + 0.337i)31-s + (−0.337 + 1.69i)37-s + (0.923 + 0.382i)43-s + (0.765 + 1.84i)49-s − 1.00·52-s + (0.962 − 1.44i)61-s + (0.707 + 0.707i)64-s + i·67-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)4-s + (−1.44 − 0.962i)7-s + (0.707 + 0.707i)13-s − 1.00i·16-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)25-s + (1.69 − 0.337i)28-s + (1.69 + 0.337i)31-s + (−0.337 + 1.69i)37-s + (0.923 + 0.382i)43-s + (0.765 + 1.84i)49-s − 1.00·52-s + (0.962 − 1.44i)61-s + (0.707 + 0.707i)64-s + i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(1.29806\)
Root analytic conductor: \(1.13932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (2377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :0),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8362557666\)
\(L(\frac12)\) \(\approx\) \(0.8362557666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (1.44 + 0.962i)T + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-1.69 - 0.337i)T + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.337 - 1.69i)T + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.923 - 0.382i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.962 + 1.44i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.962 - 1.44i)T + (-0.382 + 0.923i)T^{2} \)
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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

−9.265630551471314969373379132662, −8.416651140385196844606361708350, −7.60933007989809299924962200153, −6.86024468748873817392134280990, −6.29319306701588914936677630642, −5.09424485868720528147328380742, −4.20837018458269099910652424812, −3.51391406247809837066240119380, −2.88843492954559643082652717073, −1.01449534113811719599568728383, 0.73505683589237470744626654945, 2.32978884920030145772890947579, 3.30942820941134546129389324030, 4.14271101873147548726175724289, 5.29593921866568837192543225825, 5.94691867331736960242334932160, 6.30419294772352447641761451739, 7.48430781047481392022827605071, 8.517293256406944722313727003175, 8.963155370645885931728530371807

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