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Properties

Label 2-245-245.108-c1-0-10
Degree $2$
Conductor $245$
Sign $0.803 - 0.594i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (−0.109 + 0.0804i)2-s + (0.384 − 2.03i)3-s + (−0.584 + 1.89i)4-s + (−0.905 + 2.04i)5-s + (0.121 + 0.252i)6-s + (2.14 + 1.55i)7-s + (−0.178 − 0.509i)8-s + (−1.19 − 0.467i)9-s + (−0.0658 − 0.295i)10-s + (1.90 + 4.84i)11-s + (3.62 + 1.91i)12-s + (−0.462 − 4.10i)13-s + (−0.358 + 0.00278i)14-s + (3.80 + 2.62i)15-s + (−3.21 − 2.19i)16-s + (0.0561 + 1.50i)17-s + ⋯
L(s)  = 1  + (−0.0771 + 0.0569i)2-s + (0.222 − 1.17i)3-s + (−0.292 + 0.946i)4-s + (−0.404 + 0.914i)5-s + (0.0496 + 0.103i)6-s + (0.809 + 0.587i)7-s + (−0.0630 − 0.180i)8-s + (−0.397 − 0.155i)9-s + (−0.0208 − 0.0935i)10-s + (0.573 + 1.46i)11-s + (1.04 + 0.552i)12-s + (−0.128 − 1.13i)13-s + (−0.0958 + 0.000743i)14-s + (0.983 + 0.678i)15-s + (−0.803 − 0.547i)16-s + (0.0136 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16203 + 0.383287i\)
\(L(\frac12)\) \(\approx\) \(1.16203 + 0.383287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.905 - 2.04i)T \)
7 \( 1 + (-2.14 - 1.55i)T \)
good2 \( 1 + (0.109 - 0.0804i)T + (0.589 - 1.91i)T^{2} \)
3 \( 1 + (-0.384 + 2.03i)T + (-2.79 - 1.09i)T^{2} \)
11 \( 1 + (-1.90 - 4.84i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (0.462 + 4.10i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (-0.0561 - 1.50i)T + (-16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.977 - 1.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.87 - 0.257i)T + (22.9 + 1.71i)T^{2} \)
29 \( 1 + (-1.14 - 0.260i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (5.50 + 3.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.28 - 6.21i)T + (-20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.14 + 2.38i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (8.04 + 2.81i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (1.24 + 1.68i)T + (-13.8 + 44.9i)T^{2} \)
53 \( 1 + (-3.12 - 5.90i)T + (-29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.148 + 1.98i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (1.35 + 4.39i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (2.11 + 7.87i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.54 + 11.1i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-8.83 + 11.9i)T + (-21.5 - 69.7i)T^{2} \)
79 \( 1 + (1.36 - 0.789i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.931 + 0.104i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (-3.39 + 8.64i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (-4.04 - 4.04i)T + 97iT^{2} \)
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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

−12.28433152435081758354696634610, −11.61671088346317482356402048997, −10.35490901842952691172659285645, −8.981838679174052456981832515145, −7.86282993123855980817830596733, −7.49362363242627804656365356515, −6.57763278844391926663388219977, −4.83331555466577816358349546672, −3.31159360714895319683668461638, −2.02080326739686145491219506189, 1.16004756569571463659414967578, 3.74307837829410024434970903211, 4.66696638679520358174729165096, 5.36393039576292953579146281337, 6.98997839845986222259790999780, 8.741544468435878329018804481496, 8.944050009827554040958704834461, 9.990253713827875940417781049489, 11.16065703717213938947391331747, 11.42327522765710737042325743519

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