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Properties

Label 2-1260-28.27-c1-0-39
Degree $2$
Conductor $1260$
Sign $0.399 + 0.916i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−0.309 − 1.37i)2-s + (−1.80 + 0.853i)4-s − i·5-s + (2.64 + 0.0785i)7-s + (1.73 + 2.23i)8-s + (−1.37 + 0.309i)10-s − 0.987i·11-s + 4.69i·13-s + (−0.709 − 3.67i)14-s + (2.54 − 3.08i)16-s − 3.93i·17-s − 0.223·19-s + (0.853 + 1.80i)20-s + (−1.36 + 0.305i)22-s + 5.88i·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.975i)2-s + (−0.904 + 0.426i)4-s − 0.447i·5-s + (0.999 + 0.0296i)7-s + (0.614 + 0.789i)8-s + (−0.436 + 0.0977i)10-s − 0.297i·11-s + 1.30i·13-s + (−0.189 − 0.981i)14-s + (0.635 − 0.771i)16-s − 0.954i·17-s − 0.0513·19-s + (0.190 + 0.404i)20-s + (−0.290 + 0.0650i)22-s + 1.22i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.530143043\)
\(L(\frac12)\) \(\approx\) \(1.530143043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 1.37i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (-2.64 - 0.0785i)T \)
good11 \( 1 + 0.987iT - 11T^{2} \)
13 \( 1 - 4.69iT - 13T^{2} \)
17 \( 1 + 3.93iT - 17T^{2} \)
19 \( 1 + 0.223T + 19T^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 - 8.26T + 37T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + 4.32iT - 43T^{2} \)
47 \( 1 + 2.40T + 47T^{2} \)
53 \( 1 - 8.35T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 4.93iT - 61T^{2} \)
67 \( 1 + 7.84iT - 67T^{2} \)
71 \( 1 + 8.49iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 - 1.67T + 83T^{2} \)
89 \( 1 + 0.493iT - 89T^{2} \)
97 \( 1 + 4.31iT - 97T^{2} \)
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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

−9.462043130005301082163572610122, −8.938123535768920815603723135893, −8.114733059004189254711575107080, −7.39284191823174894036479014231, −6.07702044905927435891794123701, −4.76362202852140939377022754327, −4.55697020465057826673906240575, −3.20522166075585936457291005983, −2.04085604592119426002799405134, −1.01729481619968072130910797233, 1.00081360878366170367159832714, 2.62033409450870504969865605161, 4.08849066598894875608971896437, 4.84780358309226999783570101545, 5.79433737330847650388696201708, 6.54200169379110317412422745527, 7.47651594414680013786290711433, 8.252707324552361967326637646739, 8.571589324798018692715031201021, 9.943419379167342893274577033740

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