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Properties

Label 2-296-296.101-c1-0-12
Degree $2$
Conductor $296$
Sign $0.457 - 0.889i$
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  + (1.41 − 0.0106i)2-s + (−1.95 + 1.12i)3-s + (1.99 − 0.0301i)4-s + (−0.291 − 0.505i)5-s + (−2.74 + 1.61i)6-s + (1.42 + 2.47i)7-s + (2.82 − 0.0640i)8-s + (1.03 − 1.79i)9-s + (−0.417 − 0.711i)10-s + 2.81i·11-s + (−3.86 + 2.31i)12-s + (0.661 + 1.14i)13-s + (2.04 + 3.48i)14-s + (1.13 + 0.657i)15-s + (3.99 − 0.120i)16-s + (−1.26 − 0.728i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.00754i)2-s + (−1.12 + 0.650i)3-s + (0.999 − 0.0150i)4-s + (−0.130 − 0.225i)5-s + (−1.12 + 0.658i)6-s + (0.540 + 0.935i)7-s + (0.999 − 0.0226i)8-s + (0.345 − 0.598i)9-s + (−0.132 − 0.224i)10-s + 0.847i·11-s + (−1.11 + 0.667i)12-s + (0.183 + 0.317i)13-s + (0.547 + 0.931i)14-s + (0.293 + 0.169i)15-s + (0.999 − 0.0301i)16-s + (−0.306 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.457 - 0.889i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.457 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46880 + 0.895995i\)
\(L(\frac12)\) \(\approx\) \(1.46880 + 0.895995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0106i)T \)
37 \( 1 + (4.54 + 4.04i)T \)
good3 \( 1 + (1.95 - 1.12i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.291 + 0.505i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.42 - 2.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.81iT - 11T^{2} \)
13 \( 1 + (-0.661 - 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.26 + 0.728i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.471 - 0.816i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.03iT - 23T^{2} \)
29 \( 1 + 0.180T + 29T^{2} \)
31 \( 1 + 5.72iT - 31T^{2} \)
41 \( 1 + (3.97 + 6.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.37T + 43T^{2} \)
47 \( 1 + 7.98T + 47T^{2} \)
53 \( 1 + (-4.42 - 2.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.941 - 1.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.83 + 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.44 + 3.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.38 + 7.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + (10.2 - 5.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.78 + 1.03i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.89 - 2.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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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

−11.96907246491798608901059574228, −11.25311627982680366496563302501, −10.44844713468067670899811845560, −9.303600504902633910393151530965, −7.895154543015814878595373731438, −6.62972270618055085151562352634, −5.58742205630145985902466645375, −4.96304568399886669878658158993, −4.01965941235165034860825911019, −2.15279607777341820532056230719, 1.21935381829152804618296450313, 3.24425501364390816561983518415, 4.61150767483907923420078105529, 5.59879207136267548253398205892, 6.58794239445268531978603607081, 7.23280970920164513806729157739, 8.394841923005094278747237734295, 10.39326406701692055177160120615, 11.03134624941851006950594530636, 11.55635637211577793754657188580

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