Properties

Label 2-245-245.103-c3-0-12
Degree $2$
Conductor $245$
Sign $-0.801 + 0.598i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 + 3.93i)2-s + (−9.09 + 1.72i)3-s + (−4.70 − 15.2i)4-s + (−4.83 − 10.0i)5-s + (19.6 − 40.8i)6-s + (4.31 + 18.0i)7-s + (36.8 + 12.8i)8-s + (54.6 − 21.4i)9-s + (53.7 + 10.2i)10-s + (−8.84 + 22.5i)11-s + (69.1 + 130. i)12-s + (−39.2 − 4.42i)13-s + (−83.4 − 35.3i)14-s + (61.3 + 83.3i)15-s + (−52.3 + 35.6i)16-s + (113. + 4.23i)17-s + ⋯
L(s)  = 1  + (−1.02 + 1.39i)2-s + (−1.75 + 0.331i)3-s + (−0.588 − 1.90i)4-s + (−0.432 − 0.901i)5-s + (1.33 − 2.77i)6-s + (0.232 + 0.972i)7-s + (1.62 + 0.569i)8-s + (2.02 − 0.794i)9-s + (1.70 + 0.324i)10-s + (−0.242 + 0.617i)11-s + (1.66 + 3.14i)12-s + (−0.837 − 0.0943i)13-s + (−1.59 − 0.675i)14-s + (1.05 + 1.43i)15-s + (−0.817 + 0.557i)16-s + (1.61 + 0.0604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.801 + 0.598i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0671678 - 0.202295i\)
\(L(\frac12)\) \(\approx\) \(0.0671678 - 0.202295i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.83 + 10.0i)T \)
7 \( 1 + (-4.31 - 18.0i)T \)
good2 \( 1 + (2.90 - 3.93i)T + (-2.35 - 7.64i)T^{2} \)
3 \( 1 + (9.09 - 1.72i)T + (25.1 - 9.86i)T^{2} \)
11 \( 1 + (8.84 - 22.5i)T + (-975. - 905. i)T^{2} \)
13 \( 1 + (39.2 + 4.42i)T + (2.14e3 + 488. i)T^{2} \)
17 \( 1 + (-113. - 4.23i)T + (4.89e3 + 367. i)T^{2} \)
19 \( 1 + (-24.6 + 42.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-2.43 - 65.0i)T + (-1.21e4 + 909. i)T^{2} \)
29 \( 1 + (36.4 - 8.31i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (-166. + 96.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-320. + 169. i)T + (2.85e4 - 4.18e4i)T^{2} \)
41 \( 1 + (26.9 + 55.9i)T + (-4.29e4 + 5.38e4i)T^{2} \)
43 \( 1 + (-77.3 - 220. i)T + (-6.21e4 + 4.95e4i)T^{2} \)
47 \( 1 + (-18.6 - 13.7i)T + (3.06e4 + 9.92e4i)T^{2} \)
53 \( 1 + (577. + 305. i)T + (8.38e4 + 1.23e5i)T^{2} \)
59 \( 1 + (-8.61 - 114. i)T + (-2.03e5 + 3.06e4i)T^{2} \)
61 \( 1 + (-208. + 674. i)T + (-1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (678. + 181. i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (245. - 1.07e3i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (359. - 265. i)T + (1.14e5 - 3.71e5i)T^{2} \)
79 \( 1 + (-0.448 - 0.258i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-80.2 - 711. i)T + (-5.57e5 + 1.27e5i)T^{2} \)
89 \( 1 + (-475. - 1.21e3i)T + (-5.16e5 + 4.79e5i)T^{2} \)
97 \( 1 + (927. + 927. i)T + 9.12e5iT^{2} \)
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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

−12.09079626992513280761541019471, −11.24946511579088476704136412577, −9.788795116972776972518970956321, −9.548567146434784838112174059035, −8.064970301858160317012514355465, −7.33408260930612581837132680025, −6.03199082919917399006007531487, −5.35797769891524230058182885019, −4.71659205481469187621863568626, −1.02815243788288716859985902675, 0.22266996098466505593153704106, 1.24204777289477511799773836385, 3.10576772495140404746537741731, 4.49654236933507844360996558470, 6.05621217946758428728980230633, 7.34961182198924281194454566256, 7.927282932613712575854272658467, 9.893610306240015468793933508523, 10.39558089357988188174619606923, 10.96156383501291804997584872103

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