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Properties

Label 2-755-5.4-c1-0-53
Degree $2$
Conductor $755$
Sign $0.543 + 0.839i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.783i·2-s + 2.94i·3-s + 1.38·4-s + (−1.21 − 1.87i)5-s + 2.30·6-s − 2.34i·7-s − 2.65i·8-s − 5.66·9-s + (−1.47 + 0.952i)10-s + 3.68·11-s + 4.08i·12-s − 5.30i·13-s − 1.83·14-s + (5.52 − 3.57i)15-s + 0.694·16-s − 1.46i·17-s + ⋯
L(s)  = 1  − 0.553i·2-s + 1.69i·3-s + 0.693·4-s + (−0.543 − 0.839i)5-s + 0.941·6-s − 0.887i·7-s − 0.937i·8-s − 1.88·9-s + (−0.464 + 0.301i)10-s + 1.10·11-s + 1.17i·12-s − 1.47i·13-s − 0.491·14-s + (1.42 − 0.923i)15-s + 0.173·16-s − 0.354i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37197 - 0.746032i\)
\(L(\frac12)\) \(\approx\) \(1.37197 - 0.746032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21 + 1.87i)T \)
151 \( 1 + T \)
good2 \( 1 + 0.783iT - 2T^{2} \)
3 \( 1 - 2.94iT - 3T^{2} \)
7 \( 1 + 2.34iT - 7T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 + 5.30iT - 13T^{2} \)
17 \( 1 + 1.46iT - 17T^{2} \)
19 \( 1 + 4.99T + 19T^{2} \)
23 \( 1 + 6.83iT - 23T^{2} \)
29 \( 1 + 8.58T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 - 6.57iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 1.24iT - 43T^{2} \)
47 \( 1 - 3.49iT - 47T^{2} \)
53 \( 1 + 1.36iT - 53T^{2} \)
59 \( 1 - 5.00T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 1.06iT - 67T^{2} \)
71 \( 1 + 5.02T + 71T^{2} \)
73 \( 1 - 15.9iT - 73T^{2} \)
79 \( 1 - 2.84T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 + 5.94T + 89T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−10.28341362507832426223048080772, −9.661266275949201464338265215943, −8.718765291466413101139495973665, −7.86707193853078229156693166798, −6.61570542021040226332160026543, −5.48012505769513636013688151389, −4.24041381509944477976540223348, −3.98453044381737651872297960028, −2.82653778962239593651051807104, −0.796312388738635197760515432436, 1.78852055122684605026590668104, 2.40163058280959169318923681906, 3.87494555584778473951137071487, 5.82584237223109722144280919434, 6.31256190257174540546429625854, 7.01275716739345879734088449858, 7.53204097063189390666933151153, 8.459565058675370577065195910040, 9.260009585466749715550536760910, 10.93428720178776456296950457538

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