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Properties

Label 2-296-296.101-c1-0-23
Degree $2$
Conductor $296$
Sign $0.390 + 0.920i$
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  + (−1.36 − 0.373i)2-s + (2.16 − 1.24i)3-s + (1.72 + 1.01i)4-s + (0.890 + 1.54i)5-s + (−3.41 + 0.896i)6-s + (−1.76 − 3.05i)7-s + (−1.96 − 2.03i)8-s + (1.61 − 2.79i)9-s + (−0.639 − 2.43i)10-s − 4.13i·11-s + (4.99 + 0.0514i)12-s + (0.0436 + 0.0756i)13-s + (1.26 + 4.81i)14-s + (3.85 + 2.22i)15-s + (1.92 + 3.50i)16-s + (6.70 + 3.87i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.263i)2-s + (1.24 − 0.720i)3-s + (0.860 + 0.508i)4-s + (0.398 + 0.689i)5-s + (−1.39 + 0.365i)6-s + (−0.665 − 1.15i)7-s + (−0.696 − 0.717i)8-s + (0.538 − 0.932i)9-s + (−0.202 − 0.770i)10-s − 1.24i·11-s + (1.44 + 0.0148i)12-s + (0.0121 + 0.0209i)13-s + (0.338 + 1.28i)14-s + (0.994 + 0.574i)15-s + (0.482 + 0.876i)16-s + (1.62 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.390 + 0.920i$
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: \((2,\ 296,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06326 - 0.703943i\)
\(L(\frac12)\) \(\approx\) \(1.06326 - 0.703943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.373i)T \)
37 \( 1 + (3.42 - 5.02i)T \)
good3 \( 1 + (-2.16 + 1.24i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.890 - 1.54i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.13iT - 11T^{2} \)
13 \( 1 + (-0.0436 - 0.0756i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.70 - 3.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.34 + 4.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.36iT - 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
41 \( 1 + (-4.07 - 7.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.20T + 43T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 + (8.84 + 5.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.80 - 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.507 + 0.879i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.90 - 1.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.96 - 13.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + (-3.90 + 2.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.49 - 4.32i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.29 + 4.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.43iT - 97T^{2} \)
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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

−11.32749631584029256272191199262, −10.37964033752822265469847577037, −9.765164997985396017811712211861, −8.573175646914260062519716637342, −7.932198730142345348433853499323, −6.97860551797973278962171429918, −6.25066980320074344209498343056, −3.46220654568978310369164334744, −2.94987742280096240089975727757, −1.28317023289804414494449042420, 2.03309884545230762296605214134, 3.14645305083770280721782564598, 4.93393496044864855110914231167, 6.06492130098750903801744002321, 7.49631354787389156637928457393, 8.447698561455843425171030209759, 9.216527423474546917325592838701, 9.668986360258266682350378689765, 10.36796289631856366673706525454, 12.14015730308757147105144794720

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