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Properties

Label 2-1110-185.117-c1-0-17
Degree $2$
Conductor $1110$
Sign $0.999 + 0.0218i$
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 + (0.821 − 2.07i)5-s + (−0.707 − 0.707i)6-s + (1.82 + 1.82i)7-s + 8-s + 1.00i·9-s + (0.821 − 2.07i)10-s + 5.85i·11-s + (−0.707 − 0.707i)12-s + 2.62·13-s + (1.82 + 1.82i)14-s + (−2.05 + 0.889i)15-s + 16-s + 6.06i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.367 − 0.930i)5-s + (−0.288 − 0.288i)6-s + (0.691 + 0.691i)7-s + 0.353·8-s + 0.333i·9-s + (0.259 − 0.657i)10-s + 1.76i·11-s + (−0.204 − 0.204i)12-s + 0.726·13-s + (0.488 + 0.488i)14-s + (−0.529 + 0.229i)15-s + 0.250·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0218i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.999 + 0.0218i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.626723327\)
\(L(\frac12)\) \(\approx\) \(2.626723327\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.821 + 2.07i)T \)
37 \( 1 + (6.04 + 0.709i)T \)
good7 \( 1 + (-1.82 - 1.82i)T + 7iT^{2} \)
11 \( 1 - 5.85iT - 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 6.06iT - 17T^{2} \)
19 \( 1 + (-0.715 + 0.715i)T - 19iT^{2} \)
23 \( 1 - 5.69T + 23T^{2} \)
29 \( 1 + (0.103 + 0.103i)T + 29iT^{2} \)
31 \( 1 + (-3.13 + 3.13i)T - 31iT^{2} \)
41 \( 1 + 7.72iT - 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (0.632 + 0.632i)T + 47iT^{2} \)
53 \( 1 + (-6.74 + 6.74i)T - 53iT^{2} \)
59 \( 1 + (-2.17 + 2.17i)T - 59iT^{2} \)
61 \( 1 + (3.93 - 3.93i)T - 61iT^{2} \)
67 \( 1 + (6.21 - 6.21i)T - 67iT^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + (5.43 + 5.43i)T + 73iT^{2} \)
79 \( 1 + (4.38 - 4.38i)T - 79iT^{2} \)
83 \( 1 + (-5.79 + 5.79i)T - 83iT^{2} \)
89 \( 1 + (-2.02 - 2.02i)T + 89iT^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−9.967165679583660122449609460469, −8.850887039554782829947847303939, −8.227929295739330377893945353456, −7.17866980094073512488107328660, −6.32407440754377405054792897120, −5.35797510012166355011867843951, −4.89500696869595391685735478301, −3.90354496049899802042717407505, −2.15131409114430628901738164019, −1.51423260868319610877667905829, 1.13797803871196661152478635603, 2.96245192479624776760217173290, 3.49453686757679883142561481894, 4.75758765734617603854970941553, 5.50951216676383361854257367079, 6.39688232536163230593169585533, 7.06212145821891333616424831333, 8.096790875924903158802977334427, 9.080479280819934845840235096848, 10.18991944794661775231654490274

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