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Properties

Label 2-1110-185.117-c1-0-29
Degree $2$
Conductor $1110$
Sign $0.609 + 0.792i$
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  + 2-s + (−0.707 − 0.707i)3-s + 4-s + (2.22 + 0.215i)5-s + (−0.707 − 0.707i)6-s + (−1.35 − 1.35i)7-s + 8-s + 1.00i·9-s + (2.22 + 0.215i)10-s + 0.0194i·11-s + (−0.707 − 0.707i)12-s − 0.159·13-s + (−1.35 − 1.35i)14-s + (−1.42 − 1.72i)15-s + 16-s − 4.04i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.995 + 0.0961i)5-s + (−0.288 − 0.288i)6-s + (−0.512 − 0.512i)7-s + 0.353·8-s + 0.333i·9-s + (0.703 + 0.0679i)10-s + 0.00585i·11-s + (−0.204 − 0.204i)12-s − 0.0441·13-s + (−0.362 − 0.362i)14-s + (−0.367 − 0.445i)15-s + 0.250·16-s − 0.979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.609 + 0.792i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.514948715\)
\(L(\frac12)\) \(\approx\) \(2.514948715\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-2.22 - 0.215i)T \)
37 \( 1 + (5.84 - 1.68i)T \)
good7 \( 1 + (1.35 + 1.35i)T + 7iT^{2} \)
11 \( 1 - 0.0194iT - 11T^{2} \)
13 \( 1 + 0.159T + 13T^{2} \)
17 \( 1 + 4.04iT - 17T^{2} \)
19 \( 1 + (-4.58 + 4.58i)T - 19iT^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + (-0.812 - 0.812i)T + 29iT^{2} \)
31 \( 1 + (-2.35 + 2.35i)T - 31iT^{2} \)
41 \( 1 - 8.21iT - 41T^{2} \)
43 \( 1 - 4.36T + 43T^{2} \)
47 \( 1 + (5.16 + 5.16i)T + 47iT^{2} \)
53 \( 1 + (4.20 - 4.20i)T - 53iT^{2} \)
59 \( 1 + (-3.16 + 3.16i)T - 59iT^{2} \)
61 \( 1 + (-1.82 + 1.82i)T - 61iT^{2} \)
67 \( 1 + (-2.18 + 2.18i)T - 67iT^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + (5.46 + 5.46i)T + 73iT^{2} \)
79 \( 1 + (4.93 - 4.93i)T - 79iT^{2} \)
83 \( 1 + (-6.36 + 6.36i)T - 83iT^{2} \)
89 \( 1 + (-10.2 - 10.2i)T + 89iT^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−9.796773138714701888667030663268, −9.129441132783974747538503128315, −7.75860610942374200949709176679, −6.84709805460481937006610201306, −6.48416527737251683190725041898, −5.32427501340695701004172879714, −4.83647193547923394569818173977, −3.33352763782891122992521094409, −2.45566717222239092917850805910, −1.01643617757146017590237901787, 1.54346379619146293974796248339, 2.83289488444137300289019588842, 3.79896975805041275235433831277, 4.99040733030442994141838711583, 5.71085472370388452682003612486, 6.22401871782626195797069258810, 7.18357711097956370468009428232, 8.444292475679706453314049134227, 9.329952036533425668241875627478, 10.07988095302825838776461461941

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