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Properties

Label 1-680-680.677-r0-0-0
Degree $1$
Conductor $680$
Sign $0.0211 + 0.999i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
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  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s − 13-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 + 0.923i)23-s + (−0.923 + 0.382i)27-s + (−0.382 + 0.923i)29-s + (0.923 + 0.382i)31-s + i·33-s + (−0.382 + 0.923i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s − 13-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 + 0.923i)23-s + (−0.923 + 0.382i)27-s + (−0.382 + 0.923i)29-s + (0.923 + 0.382i)31-s + i·33-s + (−0.382 + 0.923i)37-s + (−0.382 + 0.923i)39-s + (0.382 + 0.923i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0211 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0211 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.0211 + 0.999i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.0211 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3512521343 + 0.3587673662i\)
\(L(\frac12)\) \(\approx\) \(0.3512521343 + 0.3587673662i\)
\(L(1)\) \(\approx\) \(0.7954156121 - 0.09645563613i\)
\(L(1)\) \(\approx\) \(0.7954156121 - 0.09645563613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.67985404951147916725762493760, −21.635080049471866432469852444664, −20.885010760935462387580197066, −20.19419248853102542371542365935, −19.32582635187652407243284134062, −18.69192939425278812748103748743, −17.355298612495548580961945643, −16.592225060768126106159691712324, −15.96139033615196127748527764481, −15.20027708855766406979006475023, −14.263916751715055180831667028189, −13.50171099621588754241124307844, −12.603163161557131602887416934978, −11.526127533876896398281050483901, −10.34967493862189055028818337359, −10.04423998091770664564626902975, −9.098300540582498030125999781303, −8.08461732339141952310947603196, −7.22782006539795519668481294467, −5.94784643062380734065737472262, −5.07869844872727145999369069982, −4.06261055277012499702417954355, −3.12885401536011056750083387778, −2.34323719663005940423797829698, −0.22206355710863527222543291482, 1.416954794980324572010546792560, 2.74671630859549286942094360735, 3.10068604397101733043891678902, 4.820076915537140638348613901793, 5.74519663264785694221877702505, 6.865232736424499348970686060807, 7.39691720362347927576597784592, 8.41907715440832452079449799691, 9.40896269216230999369307106430, 10.07366565046883490832445611909, 11.47052319611508002463628967310, 12.23233372996969334855446944617, 13.05056430896432468997568796936, 13.52075073164073326596279030388, 14.70579409758037596480056975937, 15.398512290279045365798816352240, 16.33074438482015032137106271542, 17.43121650095877372772877242859, 18.10018194794237541792015361743, 18.91920489979313030108728408571, 19.64168391482793245524728705410, 20.20370264327560491330988620433, 21.320085935424807579213454980885, 22.20248515761266159482042813904, 23.06678463677603823857023426199

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