Properties

Label 2-845-65.49-c1-0-30
Degree $2$
Conductor $845$
Sign $-0.968 + 0.249i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (1.33 + 2.31i)2-s + (0.416 − 0.240i)3-s + (−2.57 + 4.46i)4-s + (1.48 + 1.67i)5-s + (1.11 + 0.643i)6-s + (−0.403 + 0.698i)7-s − 8.44·8-s + (−1.38 + 2.39i)9-s + (−1.89 + 5.67i)10-s + (3.18 − 1.83i)11-s + 2.48i·12-s − 2.15·14-s + (1.02 + 0.341i)15-s + (−6.13 − 10.6i)16-s + (−1.16 − 0.675i)17-s − 7.40·18-s + ⋯
L(s)  = 1  + (0.945 + 1.63i)2-s + (0.240 − 0.138i)3-s + (−1.28 + 2.23i)4-s + (0.662 + 0.749i)5-s + (0.455 + 0.262i)6-s + (−0.152 + 0.263i)7-s − 2.98·8-s + (−0.461 + 0.799i)9-s + (−0.600 + 1.79i)10-s + (0.959 − 0.554i)11-s + 0.716i·12-s − 0.576·14-s + (0.263 + 0.0882i)15-s + (−1.53 − 2.65i)16-s + (−0.283 − 0.163i)17-s − 1.74·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326787 - 2.57967i\)
\(L(\frac12)\) \(\approx\) \(0.326787 - 2.57967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.48 - 1.67i)T \)
13 \( 1 \)
good2 \( 1 + (-1.33 - 2.31i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.416 + 0.240i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.403 - 0.698i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.18 + 1.83i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.16 + 0.675i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.45 + 0.837i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.61 + 3.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.20 - 2.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.28iT - 31T^{2} \)
37 \( 1 + (1.88 + 3.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.19 - 4.15i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.88 - 3.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.19T + 47T^{2} \)
53 \( 1 + 5.73iT - 53T^{2} \)
59 \( 1 + (5.18 + 2.99i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.884 + 1.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.94 - 8.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.41 + 4.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 - 3.84T + 83T^{2} \)
89 \( 1 + (-2.40 + 1.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.936 - 1.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.74772626904057707686563074083, −9.301432413469018080982723745825, −8.757627606838511374282368405895, −7.84385174510175008782308926486, −6.92717296961178592606800680784, −6.36231856999428485312564679680, −5.59651394064623792213234234597, −4.70559641125891939038683805700, −3.48914203755774582026221097632, −2.54610255373773414860285706214, 0.964031431324894152843183491620, 2.01611869511283718067088814669, 3.25036711295302246191184942458, 4.07589266112268800575795775030, 4.93238972933149728647461311700, 5.85220151174285790740753005008, 6.77259215399617216666955983919, 8.744954378867810853719784663875, 9.149986267487823314473511966996, 9.896547999823391729906426392198

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