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Properties

Label 2-245-7.2-c3-0-25
Degree $2$
Conductor $245$
Sign $0.991 + 0.126i$
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  + (1.15 − 2.00i)2-s + (2.5 + 4.33i)3-s + (1.31 + 2.28i)4-s + (2.5 − 4.33i)5-s + 11.5·6-s + 24.6·8-s + (0.999 − 1.73i)9-s + (−5.79 − 10.0i)10-s + (−23.1 − 40.0i)11-s + (−6.58 + 11.4i)12-s + 61.3·13-s + 25.0·15-s + (18 − 31.1i)16-s + (50.6 + 87.7i)17-s + (−2.31 − 4.01i)18-s + (1.83 − 3.17i)19-s + ⋯
L(s)  = 1  + (0.409 − 0.709i)2-s + (0.481 + 0.833i)3-s + (0.164 + 0.285i)4-s + (0.223 − 0.387i)5-s + 0.788·6-s + 1.08·8-s + (0.0370 − 0.0641i)9-s + (−0.183 − 0.317i)10-s + (−0.634 − 1.09i)11-s + (−0.158 + 0.274i)12-s + 1.30·13-s + 0.430·15-s + (0.281 − 0.487i)16-s + (0.722 + 1.25i)17-s + (−0.0303 − 0.0525i)18-s + (0.0221 − 0.0383i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.20312 - 0.203271i\)
\(L(\frac12)\) \(\approx\) \(3.20312 - 0.203271i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.15 + 2.00i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (23.1 + 40.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 61.3T + 2.19e3T^{2} \)
17 \( 1 + (-50.6 - 87.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-1.83 + 3.17i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (42.4 - 73.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 30.1T + 2.43e4T^{2} \)
31 \( 1 + (-94.4 - 163. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (9.03 - 15.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 481.T + 6.89e4T^{2} \)
43 \( 1 + 97.7T + 7.95e4T^{2} \)
47 \( 1 + (-58.8 + 101. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (333. + 578. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-28.6 - 49.6i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-369. + 639. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (276. + 478. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 740.T + 3.57e5T^{2} \)
73 \( 1 + (-116. - 202. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-537. + 931. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 683.T + 5.71e5T^{2} \)
89 \( 1 + (690. - 1.19e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 218.T + 9.12e5T^{2} \)
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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

−11.58394671254714855494840227339, −10.63556735343435195785827253546, −10.02130366768116595819410634587, −8.612561241947366107575101861945, −8.141125406824900167206485656212, −6.39369944272688583628923700509, −5.09891356320569881797998139412, −3.72519515941886934074736349436, −3.27113235036049976245479313914, −1.46433900098329622080759239298, 1.41752187347762205997632838444, 2.65213095962528783073089481806, 4.53518390951766799385481065313, 5.71209538941045048665919269835, 6.78332281927936503136572651972, 7.43698690249721245608208939815, 8.315199211438311139449602733813, 9.861989058213543284740698692682, 10.59661804451245491141750512068, 11.81268085111171700754446030783

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