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Properties

Label 2-296-296.85-c1-0-24
Degree $2$
Conductor $296$
Sign $0.319 - 0.947i$
Analytic cond. $2.36357$
Root an. cond. $1.53739$
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  + (−0.0817 + 1.41i)2-s + (2.54 + 1.46i)3-s + (−1.98 − 0.230i)4-s + (2.01 − 3.48i)5-s + (−2.28 + 3.46i)6-s + (0.477 − 0.827i)7-s + (0.488 − 2.78i)8-s + (2.80 + 4.86i)9-s + (4.75 + 3.12i)10-s + 0.266i·11-s + (−4.71 − 3.50i)12-s + (−2.19 + 3.80i)13-s + (1.12 + 0.742i)14-s + (10.2 − 5.91i)15-s + (3.89 + 0.916i)16-s + (−4.25 + 2.45i)17-s + ⋯
L(s)  = 1  + (−0.0577 + 0.998i)2-s + (1.46 + 0.847i)3-s + (−0.993 − 0.115i)4-s + (0.900 − 1.55i)5-s + (−0.930 + 1.41i)6-s + (0.180 − 0.312i)7-s + (0.172 − 0.985i)8-s + (0.936 + 1.62i)9-s + (1.50 + 0.988i)10-s + 0.0802i·11-s + (−1.36 − 1.01i)12-s + (−0.609 + 1.05i)13-s + (0.301 + 0.198i)14-s + (2.64 − 1.52i)15-s + (0.973 + 0.229i)16-s + (−1.03 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59849 + 1.14736i\)
\(L(\frac12)\) \(\approx\) \(1.59849 + 1.14736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0817 - 1.41i)T \)
37 \( 1 + (-0.468 - 6.06i)T \)
good3 \( 1 + (-2.54 - 1.46i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.01 + 3.48i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.477 + 0.827i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.266iT - 11T^{2} \)
13 \( 1 + (2.19 - 3.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.25 - 2.45i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.537 - 0.930i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.38iT - 23T^{2} \)
29 \( 1 + 9.06T + 29T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
41 \( 1 + (-2.60 + 4.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 - 9.88T + 47T^{2} \)
53 \( 1 + (-0.693 + 0.400i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.03 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.97 - 5.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 + 6.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.67 + 2.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 + (4.72 + 2.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.7 + 7.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.07 - 2.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.93iT - 97T^{2} \)
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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

−12.40181625053716824873862289696, −10.42797487261372752466066569181, −9.510949243846167760993199052270, −9.003326773469337925779238059706, −8.479483096139191278289132407230, −7.37842013588569993557453988381, −5.86977069771372393449202645146, −4.54703609918345888056472990702, −4.22439086241956664076766395887, −1.98837191199360474635983251502, 2.02033043894690312193861483767, 2.64777784479588723606839451212, 3.53638266501953771268995349390, 5.57175912481105218365298170653, 7.03395400283397085781505102524, 7.76206800519885903492184087349, 9.001727473064542176612330150166, 9.619693710215545924960434200646, 10.60760351160042910871053750578, 11.48811404033369403134433018658

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