Properties

Label 2-845-169.116-c1-0-11
Degree $2$
Conductor $845$
Sign $0.148 - 0.988i$
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  + (−1.64 − 1.85i)2-s + (1.18 + 3.12i)3-s + (−0.498 + 4.10i)4-s + (−0.239 − 0.970i)5-s + (3.84 − 7.32i)6-s + (1.44 − 1.00i)7-s + (4.35 − 3.00i)8-s + (−6.09 + 5.40i)9-s + (−1.40 + 2.03i)10-s + (0.836 − 0.943i)11-s + (−13.4 + 3.30i)12-s + (0.234 + 3.59i)13-s + (−4.23 − 1.04i)14-s + (2.74 − 1.89i)15-s + (−4.68 − 1.15i)16-s + (2.57 + 3.73i)17-s + ⋯
L(s)  = 1  + (−1.16 − 1.31i)2-s + (0.683 + 1.80i)3-s + (−0.249 + 2.05i)4-s + (−0.107 − 0.434i)5-s + (1.56 − 2.98i)6-s + (0.547 − 0.378i)7-s + (1.53 − 1.06i)8-s + (−2.03 + 1.80i)9-s + (−0.444 + 0.644i)10-s + (0.252 − 0.284i)11-s + (−3.86 + 0.953i)12-s + (0.0649 + 0.997i)13-s + (−1.13 − 0.279i)14-s + (0.709 − 0.489i)15-s + (−1.17 − 0.288i)16-s + (0.624 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652502 + 0.561947i\)
\(L(\frac12)\) \(\approx\) \(0.652502 + 0.561947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.239 + 0.970i)T \)
13 \( 1 + (-0.234 - 3.59i)T \)
good2 \( 1 + (1.64 + 1.85i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (-1.18 - 3.12i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (-1.44 + 1.00i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (-0.836 + 0.943i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (-2.57 - 3.73i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 + (-6.37 + 5.64i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (2.83 - 5.40i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (-0.997 + 1.89i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (8.10 - 3.07i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (-4.59 + 2.41i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (-8.22 + 0.999i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (-2.52 - 3.66i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (-0.856 - 3.47i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (8.64 - 12.5i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (7.22 - 0.877i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (6.99 - 2.65i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-0.227 + 0.257i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-1.60 - 13.2i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-5.38 - 2.04i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + 6.36iT - 89T^{2} \)
97 \( 1 + (-1.33 + 5.41i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.35255537194462576573612682535, −9.675107725336492113790185084792, −8.890307532883458169030411594769, −8.422610248015601937483026022552, −7.73265875645763785751803258099, −5.72521960162287404749330282890, −4.23444169999564804160452727991, −4.04085677280308490174810365997, −2.87412955132253662732136017427, −1.62907610945744624488577537638, 0.58334024735892996389872062549, 1.83901400259946106270543389250, 3.06887813411054832112950492185, 5.30050843348540375614485297729, 6.16363441613746812224827891475, 6.87388647447682078145138196745, 7.62467732523629104470159821843, 7.993404066313443873050134901633, 8.754733933369877481103309801357, 9.504600051768411395161146223688

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