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Properties

Label 2-245-7.4-c3-0-32
Degree $2$
Conductor $245$
Sign $-0.605 - 0.795i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)2-s + (−1 + 1.73i)3-s + (−0.5 + 0.866i)4-s + (2.5 + 4.33i)5-s + 6·6-s − 21·8-s + (11.5 + 19.9i)9-s + (7.50 − 12.9i)10-s + (22.5 − 38.9i)11-s + (−1.00 − 1.73i)12-s − 59·13-s − 10·15-s + (35.5 + 61.4i)16-s + (−27 + 46.7i)17-s + (34.5 − 59.7i)18-s + (−60.5 − 104. i)19-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)2-s + (−0.192 + 0.333i)3-s + (−0.0625 + 0.108i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s − 0.928·8-s + (0.425 + 0.737i)9-s + (0.237 − 0.410i)10-s + (0.616 − 1.06i)11-s + (−0.0240 − 0.0416i)12-s − 1.25·13-s − 0.172·15-s + (0.554 + 0.960i)16-s + (−0.385 + 0.667i)17-s + (0.451 − 0.782i)18-s + (−0.730 − 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-22.5 + 38.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 59T + 2.19e3T^{2} \)
17 \( 1 + (27 - 46.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (60.5 + 104. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.5 + 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 162T + 2.43e4T^{2} \)
31 \( 1 + (44 - 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-129.5 - 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 195T + 6.89e4T^{2} \)
43 \( 1 + 286T + 7.95e4T^{2} \)
47 \( 1 + (-22.5 - 38.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (298.5 - 517. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (180 - 311. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-196 - 339. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-140 + 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 48T + 3.57e5T^{2} \)
73 \( 1 + (-334 + 578. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (391 + 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 768T + 5.71e5T^{2} \)
89 \( 1 + (597 + 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 902T + 9.12e5T^{2} \)
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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

−10.92346992168105237331954304574, −10.26021769626393133744860111608, −9.381113582207653616508340074642, −8.428577478688992043387629518628, −6.95935213476336409337267378419, −5.88957691589284154573896068572, −4.52165325647870607520955010196, −2.97775683614366429770859189630, −1.80984154891676207836866886978, 0, 1.95687392609680384929042194330, 3.92838727740324194656104250926, 5.39143264305443359615586135431, 6.58319668320953836951187300449, 7.21325014567227958757266482476, 8.148237290720854196389580006949, 9.542129607324851715494398006041, 9.672811118446763062863025039888, 11.58267209736287399651128661448

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