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Properties

Label 2-296-37.26-c1-0-6
Degree $2$
Conductor $296$
Sign $0.983 + 0.180i$
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  + (−1.64 + 2.84i)3-s + (1.87 − 3.24i)5-s + (2.03 − 3.52i)7-s + (−3.90 − 6.77i)9-s + 1.66·11-s + (0.106 − 0.184i)13-s + (6.16 + 10.6i)15-s + (0.5 + 0.866i)17-s + (−0.271 + 0.470i)19-s + (6.69 + 11.6i)21-s − 4.61·23-s + (−4.51 − 7.82i)25-s + 15.8·27-s + 9.65·29-s − 1.66·31-s + ⋯
L(s)  = 1  + (−0.949 + 1.64i)3-s + (0.837 − 1.45i)5-s + (0.769 − 1.33i)7-s + (−1.30 − 2.25i)9-s + 0.503·11-s + (0.0295 − 0.0511i)13-s + (1.59 + 2.75i)15-s + (0.121 + 0.210i)17-s + (−0.0623 + 0.107i)19-s + (1.46 + 2.53i)21-s − 0.962·23-s + (−0.903 − 1.56i)25-s + 3.05·27-s + 1.79·29-s − 0.299·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12783 - 0.102556i\)
\(L(\frac12)\) \(\approx\) \(1.12783 - 0.102556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (3.22 - 5.16i)T \)
good3 \( 1 + (1.64 - 2.84i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.87 + 3.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.03 + 3.52i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 + (-0.106 + 0.184i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.271 - 0.470i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 - 9.65T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
41 \( 1 + (-0.729 + 1.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 - 3.89T + 47T^{2} \)
53 \( 1 + (-3.86 - 6.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.563 + 0.975i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.85 + 4.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.01 - 5.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.49 - 6.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 + (-4.77 + 8.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.22 - 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.99T + 97T^{2} \)
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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

−11.64289454514780989646457738394, −10.49678039833792849071029193012, −10.09427594229794734148783728481, −9.149755926111647502089947942469, −8.260014821801901100047711982238, −6.43192580154188219920422378961, −5.37618915278325509139713074930, −4.61484649399043694940939121910, −3.95755328323474478032683657618, −1.05318044361227837540908464561, 1.81586884036436769379244365660, 2.61564540665442610574924775574, 5.27138942640193230849720354291, 6.09057665585627887402551162590, 6.66764270163695705337859852451, 7.66120306060997087290318564620, 8.728779693350674513703334046595, 10.23211096865867656689422687295, 11.15555816454752241091896798644, 11.83816448979489729749279639894

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