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Properties

Label 2-1170-195.164-c1-0-20
Degree $2$
Conductor $1170$
Sign $-0.203 + 0.978i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.78 − 1.34i)5-s + (2.75 + 2.75i)7-s + (0.707 − 0.707i)8-s + (−2.21 − 0.311i)10-s + (−3.15 − 3.15i)11-s + (−2.69 − 2.39i)13-s − 3.88i·14-s − 1.00·16-s − 5.01i·17-s + (−2.53 − 2.53i)19-s + (1.34 + 1.78i)20-s + 4.45i·22-s + 4.34i·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.798 − 0.601i)5-s + (1.03 + 1.03i)7-s + (0.250 − 0.250i)8-s + (−0.700 − 0.0983i)10-s + (−0.950 − 0.950i)11-s + (−0.747 − 0.664i)13-s − 1.03i·14-s − 0.250·16-s − 1.21i·17-s + (−0.581 − 0.581i)19-s + (0.300 + 0.399i)20-s + 0.950i·22-s + 0.905i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.203 + 0.978i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.203 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.324858808\)
\(L(\frac12)\) \(\approx\) \(1.324858808\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.78 + 1.34i)T \)
13 \( 1 + (2.69 + 2.39i)T \)
good7 \( 1 + (-2.75 - 2.75i)T + 7iT^{2} \)
11 \( 1 + (3.15 + 3.15i)T + 11iT^{2} \)
17 \( 1 + 5.01iT - 17T^{2} \)
19 \( 1 + (2.53 + 2.53i)T + 19iT^{2} \)
23 \( 1 - 4.34iT - 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 + (3.60 + 3.60i)T + 31iT^{2} \)
37 \( 1 + (-4.13 - 4.13i)T + 37iT^{2} \)
41 \( 1 + (-8.67 + 8.67i)T - 41iT^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 + (-2.13 + 2.13i)T - 47iT^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + (-8.59 - 8.59i)T + 59iT^{2} \)
61 \( 1 + 2.76T + 61T^{2} \)
67 \( 1 + (4.81 - 4.81i)T - 67iT^{2} \)
71 \( 1 + (-4.04 + 4.04i)T - 71iT^{2} \)
73 \( 1 + (-0.0463 - 0.0463i)T + 73iT^{2} \)
79 \( 1 + 3.04T + 79T^{2} \)
83 \( 1 + (8.43 + 8.43i)T + 83iT^{2} \)
89 \( 1 + (-8.57 - 8.57i)T + 89iT^{2} \)
97 \( 1 + (-5.09 + 5.09i)T - 97iT^{2} \)
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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

−9.384941081802291867371924768325, −8.921415639277850216736317083686, −8.059681101394934426108869655717, −7.44930103089215538158906476753, −5.79933368853422223089060758632, −5.41621769079347832809791905654, −4.46318738252323438771939334808, −2.69702211916560950772165809512, −2.26259940314657391519902799250, −0.67740648760359321086712771429, 1.53641687931469491383209801913, 2.40311933762111553406452661937, 4.17160256227386847745680721711, 4.91812138758908861973400597146, 5.95346886584702510534139090233, 6.88742956603539323106816709631, 7.51408518662944076206109050195, 8.176274207449467890049480707756, 9.271828541368443580681573006576, 10.05957844928958610459594067833

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