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Properties

Label 2-1110-185.68-c1-0-4
Degree $2$
Conductor $1110$
Sign $-0.235 - 0.971i$
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 + (2.05 + 0.887i)5-s + (−0.707 + 0.707i)6-s + (−1.84 + 1.84i)7-s − 8-s − 1.00i·9-s + (−2.05 − 0.887i)10-s + 2.12i·11-s + (0.707 − 0.707i)12-s − 5.07·13-s + (1.84 − 1.84i)14-s + (2.07 − 0.823i)15-s + 16-s + 2.93i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.917 + 0.397i)5-s + (−0.288 + 0.288i)6-s + (−0.697 + 0.697i)7-s − 0.353·8-s − 0.333i·9-s + (−0.648 − 0.280i)10-s + 0.639i·11-s + (0.204 − 0.204i)12-s − 1.40·13-s + (0.492 − 0.492i)14-s + (0.536 − 0.212i)15-s + 0.250·16-s + 0.711i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9479276835\)
\(L(\frac12)\) \(\approx\) \(0.9479276835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 0.887i)T \)
37 \( 1 + (2.66 - 5.46i)T \)
good7 \( 1 + (1.84 - 1.84i)T - 7iT^{2} \)
11 \( 1 - 2.12iT - 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 2.93iT - 17T^{2} \)
19 \( 1 + (-2.28 - 2.28i)T + 19iT^{2} \)
23 \( 1 + 5.76T + 23T^{2} \)
29 \( 1 + (5.76 - 5.76i)T - 29iT^{2} \)
31 \( 1 + (-1.38 - 1.38i)T + 31iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 2.00T + 43T^{2} \)
47 \( 1 + (-7.27 + 7.27i)T - 47iT^{2} \)
53 \( 1 + (0.286 + 0.286i)T + 53iT^{2} \)
59 \( 1 + (8.25 + 8.25i)T + 59iT^{2} \)
61 \( 1 + (-2.44 - 2.44i)T + 61iT^{2} \)
67 \( 1 + (-9.89 - 9.89i)T + 67iT^{2} \)
71 \( 1 - 1.69T + 71T^{2} \)
73 \( 1 + (5.92 - 5.92i)T - 73iT^{2} \)
79 \( 1 + (-5.73 - 5.73i)T + 79iT^{2} \)
83 \( 1 + (7.41 + 7.41i)T + 83iT^{2} \)
89 \( 1 + (-11.9 + 11.9i)T - 89iT^{2} \)
97 \( 1 - 7.94iT - 97T^{2} \)
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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

−9.958110417800352027016596019530, −9.354444457973704191231118668477, −8.590745195304946277695559714830, −7.52175690493921399298849452631, −6.91701633327605467832781778457, −6.04509758889995246202078611766, −5.23362285759735013188081845319, −3.52610707455326835001055967147, −2.44435521635500627605710467853, −1.78834666083363936117602702589, 0.45718737311059567746623835997, 2.13448684970749194139582045975, 3.05386677126182535834057975586, 4.32539216536898971151701436195, 5.40815686605344455028581989909, 6.30372932718140813304244044812, 7.31378970668977104833156950630, 7.977565596065890224039014796827, 9.160358695779743707553763867010, 9.573720112987243795190023279582

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