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Properties

Label 2-1110-37.11-c1-0-17
Degree $2$
Conductor $1110$
Sign $0.367 + 0.929i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·6-s + (1.36 − 2.36i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 3.73·11-s + (−0.499 − 0.866i)12-s + (1.5 + 0.866i)13-s + 2.73i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (2.76 − 1.59i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + 0.408i·6-s + (0.516 − 0.894i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 1.12·11-s + (−0.144 − 0.249i)12-s + (0.416 + 0.240i)13-s + 0.730i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.671 − 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358526852\)
\(L(\frac12)\) \(\approx\) \(1.358526852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.5 + 6.06i)T \)
good7 \( 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.73T + 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.76 + 1.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.73 + i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 7.73iT - 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
41 \( 1 + (-1.09 + 1.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + (1.09 + 1.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.59 - 3.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.09 + 2.36i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.33 + 5.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.09 + 1.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.39T + 73T^{2} \)
79 \( 1 + (7.73 + 4.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.09 + 3.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.803 + 0.464i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.53iT - 97T^{2} \)
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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

−9.357861765097778197725027098154, −8.886835049209957948837246555800, −7.922603355247736158866583325854, −7.32985936445786624088387442263, −6.65700620800205736714713662073, −5.59985658519175922293914364867, −4.37054812553548779747442319228, −3.48808343248662938342326992923, −1.80562580875420734450644803966, −0.806240403533146537900804973920, 1.42172925744207286899241271191, 2.69500991879148434362156197921, 3.67615239226642480937231059758, 4.58921834690005952464698965910, 5.84692777697462206033659513840, 6.69698291439977661009480293055, 7.975211584543687302255208094413, 8.390077523756719231037340024146, 9.165821053492425174576416871484, 9.936495539050099498329526510384

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