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Properties

Label 2-296-296.101-c1-0-25
Degree $2$
Conductor $296$
Sign $0.946 + 0.322i$
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  + (1.38 − 0.307i)2-s + (0.600 − 0.346i)3-s + (1.81 − 0.849i)4-s + (1.31 + 2.26i)5-s + (0.721 − 0.663i)6-s + (−1.74 − 3.02i)7-s + (2.23 − 1.72i)8-s + (−1.25 + 2.18i)9-s + (2.50 + 2.72i)10-s − 1.40i·11-s + (0.792 − 1.13i)12-s + (2.32 + 4.01i)13-s + (−3.33 − 3.63i)14-s + (1.57 + 0.908i)15-s + (2.55 − 3.07i)16-s + (−5.28 − 3.05i)17-s + ⋯
L(s)  = 1  + (0.976 − 0.217i)2-s + (0.346 − 0.200i)3-s + (0.905 − 0.424i)4-s + (0.586 + 1.01i)5-s + (0.294 − 0.270i)6-s + (−0.659 − 1.14i)7-s + (0.791 − 0.611i)8-s + (−0.419 + 0.727i)9-s + (0.792 + 0.863i)10-s − 0.423i·11-s + (0.228 − 0.328i)12-s + (0.643 + 1.11i)13-s + (−0.891 − 0.970i)14-s + (0.406 + 0.234i)15-s + (0.639 − 0.768i)16-s + (−1.28 − 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.946 + 0.322i$
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: \((2,\ 296,\ (\ :1/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48063 - 0.410407i\)
\(L(\frac12)\) \(\approx\) \(2.48063 - 0.410407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.307i)T \)
37 \( 1 + (-5.70 + 2.11i)T \)
good3 \( 1 + (-0.600 + 0.346i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.31 - 2.26i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.74 + 3.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.40iT - 11T^{2} \)
13 \( 1 + (-2.32 - 4.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.28 + 3.05i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 + 3.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.02iT - 23T^{2} \)
29 \( 1 + 6.17T + 29T^{2} \)
31 \( 1 - 1.36iT - 31T^{2} \)
41 \( 1 + (3.13 + 5.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.90T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 + (-2.05 - 1.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.01 - 1.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.65 - 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.98 - 2.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.65 - 6.33i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + (-7.01 + 4.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.75 + 5.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.2 - 8.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.77iT - 97T^{2} \)
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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

−11.36369881401829730877264854868, −11.10136547361690068201498318678, −10.14925088000555474356199466593, −9.029804402486336708902483711644, −7.29114586535581238025060646214, −6.83274460223451864176893927996, −5.81758027992261027390682364951, −4.31782780243944193707275534438, −3.20046093370150651325599838647, −2.08708520286825404028590084923, 2.18428420073383546534032381715, 3.45483787091659412040816520353, 4.74114983576310228373047573269, 5.93493460187933471299012905661, 6.31876939747982042737523488771, 8.208321342100222656635078606245, 8.816258393565583636139253739135, 9.820520229351182290515469599168, 11.09110060109752651971752475451, 12.25378113717474969969874113580

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