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Properties

Label 2-2720-85.2-c0-0-1
Degree $2$
Conductor $2720$
Sign $0.485 + 0.874i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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  − 5-s + (0.707 + 0.707i)9-s + (−1.41 − 1.41i)13-s − i·17-s + 25-s + (1.70 − 0.707i)29-s + (−0.707 − 0.292i)37-s + (0.707 − 1.70i)41-s + (−0.707 − 0.707i)45-s + (−0.707 + 0.707i)49-s + 2·53-s + (0.292 − 0.707i)61-s + (1.41 + 1.41i)65-s + (−0.707 − 0.292i)73-s + 1.00i·81-s + ⋯
L(s)  = 1  − 5-s + (0.707 + 0.707i)9-s + (−1.41 − 1.41i)13-s − i·17-s + 25-s + (1.70 − 0.707i)29-s + (−0.707 − 0.292i)37-s + (0.707 − 1.70i)41-s + (−0.707 − 0.707i)45-s + (−0.707 + 0.707i)49-s + 2·53-s + (0.292 − 0.707i)61-s + (1.41 + 1.41i)65-s + (−0.707 − 0.292i)73-s + 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 0.485 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8883163843\)
\(L(\frac12)\) \(\approx\) \(0.8883163843\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.722420698607616026388977451556, −8.000338506581122850475748002374, −7.36466051324531273708460294943, −6.98952024841670398662781807929, −5.60345929453673047027248157024, −4.88265255485271481630236475307, −4.27713202818517893082327210097, −3.11584753542377717244738252194, −2.36494800697101746904166178124, −0.63293400464338781877074227959, 1.28881433105718145698145379969, 2.59246486607837320568654330098, 3.69710976540952117420044832179, 4.37908409587494497838246730407, 4.97781969579118755950067936075, 6.38924181066208914516467381354, 6.88601791224466732470424168499, 7.52425587527988150741834244515, 8.440770258831279094905349284827, 9.035634741323337758458389713749

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